From e45871795b26417b8b43706bff8487a6cd8fc1da Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 19 Nov 2020 05:48:03 +0100 Subject: [PATCH] svm update --- doc/pub/week47/html/._week47-bs000.html | 76 +++-- doc/pub/week47/html/._week47-bs001.html | 76 +++-- doc/pub/week47/html/._week47-bs002.html | 76 +++-- doc/pub/week47/html/._week47-bs003.html | 76 +++-- doc/pub/week47/html/._week47-bs004.html | 76 +++-- doc/pub/week47/html/._week47-bs005.html | 76 +++-- doc/pub/week47/html/._week47-bs006.html | 76 +++-- doc/pub/week47/html/._week47-bs007.html | 76 +++-- doc/pub/week47/html/._week47-bs008.html | 76 +++-- doc/pub/week47/html/._week47-bs009.html | 76 +++-- doc/pub/week47/html/._week47-bs010.html | 76 +++-- doc/pub/week47/html/._week47-bs011.html | 76 +++-- doc/pub/week47/html/._week47-bs012.html | 76 +++-- doc/pub/week47/html/._week47-bs013.html | 100 ++++--- doc/pub/week47/html/._week47-bs014.html | 118 +++++--- doc/pub/week47/html/._week47-bs015.html | 134 ++++----- doc/pub/week47/html/._week47-bs016.html | 121 ++++---- doc/pub/week47/html/._week47-bs017.html | 122 ++++---- doc/pub/week47/html/._week47-bs018.html | 115 ++++---- doc/pub/week47/html/._week47-bs019.html | 110 +++---- doc/pub/week47/html/._week47-bs020.html | 115 ++++---- doc/pub/week47/html/._week47-bs021.html | 135 +++++---- doc/pub/week47/html/._week47-bs022.html | 174 ++++++----- doc/pub/week47/html/._week47-bs023.html | 169 +++++------ doc/pub/week47/html/._week47-bs024.html | 138 ++++----- doc/pub/week47/html/._week47-bs025.html | 131 +++++---- doc/pub/week47/html/._week47-bs026.html | 287 ++++++++++++++----- doc/pub/week47/html/._week47-bs027.html | 280 ++++-------------- doc/pub/week47/html/._week47-bs028.html | 106 ++++--- doc/pub/week47/html/._week47-bs029.html | 145 ++++++---- doc/pub/week47/html/._week47-bs030.html | 147 ++++------ doc/pub/week47/html/week47-bs.html | 76 +++-- doc/pub/week47/html/week47-reveal.html | 55 ++-- doc/pub/week47/html/week47-solarized.html | 91 +++--- doc/pub/week47/html/week47.html | 91 +++--- doc/pub/week47/ipynb/ipynb-week47-src.tar.gz | Bin 190 -> 191 bytes doc/pub/week47/ipynb/week47.ipynb | 18 +- doc/src/week47/week47.do.txt | 17 +- 38 files changed, 1947 insertions(+), 2036 deletions(-) diff --git a/doc/pub/week47/html/._week47-bs000.html b/doc/pub/week47/html/._week47-bs000.html index e96c3a089..f6f7353f5 100644 --- a/doc/pub/week47/html/._week47-bs000.html +++ b/doc/pub/week47/html/._week47-bs000.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
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  • Solving the equations
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  • Code Example
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  • Problems with the Simpler Approach
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  • A better approach
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  • A quick Reminder on Lagrangian Multipliers
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  • Adding the Multiplier
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  • Setting up the Problem
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  • The problem to solve
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  • The last steps
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  • Mathematical optimization of convex functions
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  • How do we solve these problems?
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  • A simple example
    • -
  • Back to the more realistic cases
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  • Can we code this?
    • +
  • A better approach
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  • A quick Reminder on Lagrangian Multipliers
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  • Adding the Multiplier
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  • Setting up the Problem
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  • The problem to solve
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  • The last steps
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  • A soft classifier
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  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
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  • Different kernels and Mercer's theorem
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  • The moons example
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  • Mathematical optimization of convex functions
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  • How do we solve these problems?
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  • A simple example
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  • Back to the more realistic cases
    • @@ -204,7 +202,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week47/html/._week47-bs001.html b/doc/pub/week47/html/._week47-bs001.html index 91fe9b148..5ad61894f 100644 --- a/doc/pub/week47/html/._week47-bs001.html +++ b/doc/pub/week47/html/._week47-bs001.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
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  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
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  • Setting up the Problem
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  • The problem to solve
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  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -190,7 +188,7 @@ Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) o
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    • diff --git a/doc/pub/week47/html/._week47-bs002.html b/doc/pub/week47/html/._week47-bs002.html index dfb2ca5de..76db46a10 100644 --- a/doc/pub/week47/html/._week47-bs002.html +++ b/doc/pub/week47/html/._week47-bs002.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
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  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -184,7 +182,7 @@ We start with our final topic this semester, Support Vector Machines
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    • diff --git a/doc/pub/week47/html/._week47-bs003.html b/doc/pub/week47/html/._week47-bs003.html index 6e19a08f3..4ab75b994 100644 --- a/doc/pub/week47/html/._week47-bs003.html +++ b/doc/pub/week47/html/._week47-bs003.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
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  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
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  • A quick Reminder on Lagrangian Multipliers
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  • Adding the Multiplier
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  • Setting up the Problem
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  • The problem to solve
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  • The last steps
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  • A soft classifier
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  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -185,7 +183,7 @@ Friday's lecture is split in two parts. The first lecture is deveoted to a prese
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    • diff --git a/doc/pub/week47/html/._week47-bs004.html b/doc/pub/week47/html/._week47-bs004.html index b49fab415..ae63e1737 100644 --- a/doc/pub/week47/html/._week47-bs004.html +++ b/doc/pub/week47/html/._week47-bs004.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
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  • A soft classifier
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  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
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  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -196,7 +194,7 @@ Here are the various projects that will be presented during the first lecture (a
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    • diff --git a/doc/pub/week47/html/._week47-bs005.html b/doc/pub/week47/html/._week47-bs005.html index caac38707..3becf03de 100644 --- a/doc/pub/week47/html/._week47-bs005.html +++ b/doc/pub/week47/html/._week47-bs005.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -212,7 +210,7 @@ unlikely that we can separate classes easily by say straight lines.
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    • -
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    • diff --git a/doc/pub/week47/html/._week47-bs006.html b/doc/pub/week47/html/._week47-bs006.html index 8460a426e..eb9473a12 100644 --- a/doc/pub/week47/html/._week47-bs006.html +++ b/doc/pub/week47/html/._week47-bs006.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -267,7 +265,7 @@ plt.show()
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week47/html/._week47-bs007.html b/doc/pub/week47/html/._week47-bs007.html index c8f856dc1..3fcd27301 100644 --- a/doc/pub/week47/html/._week47-bs007.html +++ b/doc/pub/week47/html/._week47-bs007.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -212,7 +210,7 @@ $$
  • 16
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week47/html/._week47-bs008.html b/doc/pub/week47/html/._week47-bs008.html index 348dc4b0c..88a771945 100644 --- a/doc/pub/week47/html/._week47-bs008.html +++ b/doc/pub/week47/html/._week47-bs008.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -224,7 +222,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
  • 17
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    • -
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    • diff --git a/doc/pub/week47/html/._week47-bs009.html b/doc/pub/week47/html/._week47-bs009.html index f1c2ecb0e..8375de71d 100644 --- a/doc/pub/week47/html/._week47-bs009.html +++ b/doc/pub/week47/html/._week47-bs009.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -210,7 +208,7 @@ for our data sample.
  • 18
  • 19
  • ...
    • -
  • 32
    • +
  • 31
  • »
    • diff --git a/doc/pub/week47/html/._week47-bs010.html b/doc/pub/week47/html/._week47-bs010.html index 3027ccbfa..c05fb0d4f 100644 --- a/doc/pub/week47/html/._week47-bs010.html +++ b/doc/pub/week47/html/._week47-bs010.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
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  • Kernels and non-linearity
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  • The equations
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  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
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  • A better approach
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  • A quick Reminder on Lagrangian Multipliers
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  • Adding the Multiplier
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  • Setting up the Problem
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  • The problem to solve
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  • The last steps
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  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -206,7 +204,7 @@ $$
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    • -
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    • +
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  • »
    • diff --git a/doc/pub/week47/html/._week47-bs011.html b/doc/pub/week47/html/._week47-bs011.html index a8da297a2..3a8197e79 100644 --- a/doc/pub/week47/html/._week47-bs011.html +++ b/doc/pub/week47/html/._week47-bs011.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -209,7 +207,7 @@ $$
  • 20
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    • -
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    • +
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  • »
    • diff --git a/doc/pub/week47/html/._week47-bs012.html b/doc/pub/week47/html/._week47-bs012.html index 5efebf7b3..64e370743 100644 --- a/doc/pub/week47/html/._week47-bs012.html +++ b/doc/pub/week47/html/._week47-bs012.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -202,7 +200,7 @@ where \( \eta \) is our by now well-known learning rate.
  • 21
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    • diff --git a/doc/pub/week47/html/._week47-bs013.html b/doc/pub/week47/html/._week47-bs013.html index f4f20dafc..c44f2ad49 100644 --- a/doc/pub/week47/html/._week47-bs013.html +++ b/doc/pub/week47/html/._week47-bs013.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,17 +159,27 @@ MathJax.Hub.Config({ -

    Code Example

    +

    Can we code this?

    The equations we discussed above can be coded rather easily (the -framework is similar to what we developed for logistic -regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way. -

    +framework is similar to what we developed for logistic regression). We +can set up a simple case with two classes only and we want to find a +line which separates them the best possible way. + +

    +There are however problems with this approach, although it looks +pretty straightforward to implement. When running a code for such a +case we can easily end up with many diffeent lines which separate the +two classes. + +

    +For small +gaps between the entries, we may also end up needing many iterations +before the solutions converge and if the data cannot be separated +properly into two distinct classes, we may not experience a converge +at all. - -

    
-

    @@ -198,7 +206,7 @@ regression). We are going to set up a simple case with two classes only and we w

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    • diff --git a/doc/pub/week47/html/._week47-bs014.html b/doc/pub/week47/html/._week47-bs014.html index 5ae161f80..2fbfa74c4 100644 --- a/doc/pub/week47/html/._week47-bs014.html +++ b/doc/pub/week47/html/._week47-bs014.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,18 +159,44 @@ MathJax.Hub.Config({ -

    Problems with the Simpler Approach

    +

    A better approach

    -There are however problems with this approach, although it looks -pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes. +A better approach is rather to try to define a large margin between +the two classes (if they are well separated from the beginning).

    -For small -gaps between the entries, we may also end up needing many iterations -before the solutions converge and if the data cannot be separated -properly into two distinct classes, we may not experience a converge -at all. +Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to +\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition + +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. +$$ + +All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. + +

    +We seek thus the largest value \( M \) defined by +$$ +\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, +$$ + +or just +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. +$$ + +If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of +\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. +$$ + +

    +We have thus defined our margin as the invers of the norm of +\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as +possible margin \( M \). Before we proceed, we need to remind ourselves +about Lagrangian multipliers.

    @@ -200,7 +224,7 @@ at all.

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    • diff --git a/doc/pub/week47/html/._week47-bs015.html b/doc/pub/week47/html/._week47-bs015.html index 884dee6de..35a39f87b 100644 --- a/doc/pub/week47/html/._week47-bs015.html +++ b/doc/pub/week47/html/._week47-bs015.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,44 +159,52 @@ MathJax.Hub.Config({ -

    A better approach

    +

    A quick Reminder on Lagrangian Multipliers

    -A better approach is rather to try to define a large margin between -the two classes (if they are well separated from the beginning). +Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an +extreme we have +$$ +df=0. +$$ + +A necessary and sufficient condition is +$$ +\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, +$$ + +due to +$$ +df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. +$$ + +In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin) +so that they are no longer all independent. It is possible at least in principle to use each +constraint to eliminate one variable +and to proceed with a new and smaller set of independent varables.

    -Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to -\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition - +The use of so-called Lagrangian multipliers is an alternative technique when the elimination +of variables is incovenient or undesirable. Assume that we have an equation of constraint on +the variables \( x,y,z \) $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. +\phi(x,y,z) = 0, $$ -All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. - -

    -We seek thus the largest value \( M \) defined by + resulting in $$ -\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, +d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. $$ -or just +Now we cannot set anymore $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. +\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, $$ -If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of -\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. -$$ - -

    -We have thus defined our margin as the invers of the norm of -\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as -possible margin \( M \). Before we proceed, we need to remind ourselves -about Lagrangian multipliers. +if \( df=0 \) is wanted +because there are now only two independent variables! Assume \( x \) and \( y \) are the independent +variables. +Then \( dz \) is no longer arbitrary.

    @@ -226,7 +232,7 @@ about Lagrangian multipliers.

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    • diff --git a/doc/pub/week47/html/._week47-bs016.html b/doc/pub/week47/html/._week47-bs016.html index 08c0a0ca8..e74b3a50f 100644 --- a/doc/pub/week47/html/._week47-bs016.html +++ b/doc/pub/week47/html/._week47-bs016.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,53 +159,46 @@ MathJax.Hub.Config({ -

    A quick Reminder on Lagrangian Multipliers

    +

    Adding the Multiplier

    -Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an -extreme we have +However, we can add to $$ -df=0. +df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, $$ -A necessary and sufficient condition is +a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in $$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, +df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda +\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ +(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. $$ -due to +Our multiplier is chosen so that $$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. +\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. $$ -In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin) -so that they are no longer all independent. It is possible at least in principle to use each -constraint to eliminate one variable -and to proceed with a new and smaller set of independent varables. -

    -The use of so-called Lagrangian multipliers is an alternative technique when the elimination -of variables is incovenient or undesirable. Assume that we have an equation of constraint on -the variables \( x,y,z \) +We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have $$ -\phi(x,y,z) = 0, +\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, $$ - resulting in +and $$ -d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. +\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. $$ -Now we cannot set anymore +When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and +\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined, +it is therefore often called +Lagrange's undetermined multiplier. +If we have a set of constraints \( \phi_k \) we have the equations $$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, +\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. $$ -if \( df=0 \) is wanted -because there are now only two independent variables! Assume \( x \) and \( y \) are the independent -variables. -Then \( dz \) is no longer arbitrary. -

    @@ -234,7 +225,7 @@ Then \( dz \) is no longer arbitrary.

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    • diff --git a/doc/pub/week47/html/._week47-bs017.html b/doc/pub/week47/html/._week47-bs017.html index 93c641fba..93f91e58f 100644 --- a/doc/pub/week47/html/._week47-bs017.html +++ b/doc/pub/week47/html/._week47-bs017.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,45 +159,43 @@ MathJax.Hub.Config({ -

    Adding the Multiplier

    +

    Setting up the Problem

    +In order to solve the above problem, we define the following Lagrangian function to be minimized +$$ +{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], +$$ + +where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \).

    -However, we can add to +Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain $$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, +\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, $$ -a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in +and $$ -df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda -\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ -(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. +\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. $$ -Our multiplier is chosen so that +Inserting these constraints into the equation for \( {\cal L} \) we obtain $$ -\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, $$ -

    -We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have +subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). +We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition $$ -\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, +\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. $$ -and -$$ -\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. -$$ -When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and -\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined, -it is therefore often called -Lagrange's undetermined multiplier. -If we have a set of constraints \( \phi_k \) we have the equations -$$ -\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. -$$ +

      +
    1. If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.
      2. +
    2. If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
      3. +
    + +When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \).

    @@ -227,7 +223,7 @@ $$

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    • diff --git a/doc/pub/week47/html/._week47-bs018.html b/doc/pub/week47/html/._week47-bs018.html index f6fd45b55..e7ffc11a8 100644 --- a/doc/pub/week47/html/._week47-bs018.html +++ b/doc/pub/week47/html/._week47-bs018.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,43 +159,26 @@ MathJax.Hub.Config({ -

    Setting up the Problem

    -In order to solve the above problem, we define the following Lagrangian function to be minimized -$$ -{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], -$$ - -where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \). +

    The problem to solve

    -Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain -$$ -\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, -$$ - -and -$$ -\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. -$$ - -Inserting these constraints into the equation for \( {\cal L} \) we obtain +We can rewrite $$ {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, $$ -subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). -We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition +and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem $$ -\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. +\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ +y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ +\dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots \\ +y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ +\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, $$ - -

      -
    1. If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.
      2. -
    2. If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
      3. -
    - -When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \). +subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and +\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).

    @@ -225,7 +206,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support

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    • diff --git a/doc/pub/week47/html/._week47-bs019.html b/doc/pub/week47/html/._week47-bs019.html index 43a03c61f..7d24fb9f2 100644 --- a/doc/pub/week47/html/._week47-bs019.html +++ b/doc/pub/week47/html/._week47-bs019.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,26 +159,36 @@ MathJax.Hub.Config({ -

    The problem to solve

    +

    The last steps

    -We can rewrite +Solving the above problem, yields the values of \( \lambda_i \). +To find the coefficients of your hyperplane we need simply to compute $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, +\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. $$ -and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem +With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via $$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ -y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, $$ -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). +resulting in +$$ +b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, +$$ + +or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have +$$ +b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). +$$ + +With our hyperplane coefficients we can use our classifier to assign any observation by simply using +$$ +y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). +$$ + +Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier.

    @@ -208,7 +216,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec

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    • diff --git a/doc/pub/week47/html/._week47-bs020.html b/doc/pub/week47/html/._week47-bs020.html index 6ec6015e8..973a3b8c8 100644 --- a/doc/pub/week47/html/._week47-bs020.html +++ b/doc/pub/week47/html/._week47-bs020.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,36 +159,37 @@ MathJax.Hub.Config({ -

    The last steps

    +

    A soft classifier

    -Solving the above problem, yields the values of \( \lambda_i \). -To find the coefficients of your hyperplane we need simply to compute -$$ -\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. -$$ +Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined. -With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via +

    +Suppose now that classes overlap in feature space, as shown in the +figure here. One way to deal with this problem before we define the +so-called kernel approach, is to allow a kind of slack in the sense +that we allow some points to be on the wrong side of the margin. + +

    +We introduce thus the so-called slack variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and +modify our previous equation $$ y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, $$ -resulting in +to $$ -b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, $$ -or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have -$$ -b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). -$$ +with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \). +The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction +\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \), +we bound the total amount by which predictions fall on the wrong side of their margins. -With our hyperplane coefficients we can use our classifier to assign any observation by simply using -$$ -y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). -$$ - -Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier. +

    +Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of +misclassifications.

    @@ -218,7 +217,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr

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    • diff --git a/doc/pub/week47/html/._week47-bs021.html b/doc/pub/week47/html/._week47-bs021.html index a7de379d9..bf1e412a3 100644 --- a/doc/pub/week47/html/._week47-bs021.html +++ b/doc/pub/week47/html/._week47-bs021.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,37 +159,56 @@ MathJax.Hub.Config({ -

    A soft classifier

    +

    Soft optmization problem

    -Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined. +This has in turn the consequences that we change our optmization problem to finding the minimum of +$$ +{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, +$$ + +subject to +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, +$$ + +with the requirement \( \xi_i\geq 0 \).

    -Suppose now that classes overlap in feature space, as shown in the -figure here. One way to deal with this problem before we define the -so-called kernel approach, is to allow a kind of slack in the sense -that we allow some points to be on the wrong side of the margin. - -

    -We introduce thus the so-called slack variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and -modify our previous equation +Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, +\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, $$ -to +and $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, +\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, $$ -with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \). -The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction -\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \), -we bound the total amount by which predictions fall on the wrong side of their margins. +and +$$ +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. +$$ -

    -Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of -misclassifications. +Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before +$$ +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, +$$ + +but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \). +We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads +$$ +\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, +$$ + +$$ +\gamma_i\xi_i = 0, +$$ + +and +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. +$$

    @@ -218,8 +235,6 @@ misclassifications.

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    • diff --git a/doc/pub/week47/html/._week47-bs022.html b/doc/pub/week47/html/._week47-bs022.html index e22e4cf7a..ba925b75a 100644 --- a/doc/pub/week47/html/._week47-bs022.html +++ b/doc/pub/week47/html/._week47-bs022.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,57 +159,76 @@ MathJax.Hub.Config({ -

    Soft optmization problem

    +

    Kernels and non-linearity

    -This has in turn the consequences that we change our optmization problem to finding the minimum of -$$ -{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, -$$ - -subject to -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, -$$ - -with the requirement \( \xi_i\geq 0 \). +The cases we have studied till now, were all characterized by two classes +with a close to linear separability. The classifiers we have described +so far find linear boundaries in our input feature space. It is +possible to make our procedure more flexible by exploring the feature +space using other basis expansions such as higher-order polynomials, +wavelets, splines etc.

    -Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain -$$ -\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, -$$ +If our feature space is not easy to separate, as shown in the figure +here, we can achieve a better separation by introducing more complex +basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to +obtain a separation between the classes which is almost linear. -and -$$ -\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, -$$ +

    +The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that +we need to introduce for example a polynomial transformation to a two-dimensional training set. -and -$$ -\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. -$$ +

    -Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -$$ + +

    import numpy as np
+import os
 
-but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \). 
-We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads
-$$
-\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i,
-$$
+np.random.seed(42)
 
-$$
-\gamma_i\xi_i = 0,
-$$
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
 
-and 
-$$
-y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i.
-$$
 
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
+X2D = np.c_[X1D, X1D**2]
+y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
+plt.gca().get_yaxis().set_ticks([])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.axis([-4.5, 4.5, -0.2, 0.2])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
+plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
+plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
+plt.axis([-4.5, 4.5, -1, 17])
+plt.subplots_adjust(right=1)
+plt.show()
+

    @@ -236,7 +253,6 @@ $$

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  • »
    • diff --git a/doc/pub/week47/html/._week47-bs023.html b/doc/pub/week47/html/._week47-bs023.html index 67433da6e..d1f76b67f 100644 --- a/doc/pub/week47/html/._week47-bs023.html +++ b/doc/pub/week47/html/._week47-bs023.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,76 +159,48 @@ MathJax.Hub.Config({ -

    Kernels and non-linearity

    +

    The equations

    -The cases we have studied till now, were all characterized by two classes -with a close to linear separability. The classifiers we have described -so far find linear boundaries in our input feature space. It is -possible to make our procedure more flexible by exploring the feature -space using other basis expansions such as higher-order polynomials, -wavelets, splines etc. +Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) +$$ +z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). +$$

    -If our feature space is not easy to separate, as shown in the figure -here, we can achieve a better separation by introducing more complex -basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to -obtain a separation between the classes which is almost linear. +With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) +$$ +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, +$$ + +subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors +$$ +y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, +$$ + +from which we also find \( b \). +To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +$$ +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). +$$ + +For the above example, the kernel reads +$$ +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. +$$

    -The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that -we need to introduce for example a polynomial transformation to a two-dimensional training set. +We note that this is nothing but the dot product of the two original +vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the +product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute +the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \).

    +This leads to the so-called +kernel trick and the result leads to the same as if we went through +the trouble of performing the transformation +\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. - -

    import numpy as np
-import os
-
-np.random.seed(42)
-
-# To plot pretty figures
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-from sklearn.svm import SVC
-from sklearn import datasets
-
-
-
-X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
-X2D = np.c_[X1D, X1D**2]
-y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
-plt.gca().get_yaxis().set_ticks([])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.axis([-4.5, 4.5, -0.2, 0.2])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
-plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
-plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
-plt.axis([-4.5, 4.5, -1, 17])
-plt.subplots_adjust(right=1)
-plt.show()
-

    @@ -254,7 +224,6 @@ plt.show()

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    • diff --git a/doc/pub/week47/html/._week47-bs024.html b/doc/pub/week47/html/._week47-bs024.html index b33a5f4b5..2f88196bb 100644 --- a/doc/pub/week47/html/._week47-bs024.html +++ b/doc/pub/week47/html/._week47-bs024.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,47 +159,38 @@ MathJax.Hub.Config({ -

    The equations

    +

    The problem to solve

    +Using our definition of the kernel We can rewrite again the Lagrangian +$$ +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, +$$ + +subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem +$$ +\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ +y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ +\dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots \\ +y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ +\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, +$$ + +subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and +\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). +If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).

    -Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) +We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type $$ -z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). +\begin{align*} + &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber + &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. +\end{align*} $$ -

    -With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, -$$ - -subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors -$$ -y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, -$$ - -from which we also find \( b \). -To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as -$$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). -$$ - -For the above example, the kernel reads -$$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. -$$ - -

    -We note that this is nothing but the dot product of the two original -vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the -product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute -the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). - -

    -This leads to the so-called -kernel trick and the result leads to the same as if we went through -the trouble of performing the transformation -\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. +Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \). +Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into +\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \).

    @@ -225,7 +214,6 @@ the trouble of performing the transformation

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    • diff --git a/doc/pub/week47/html/._week47-bs025.html b/doc/pub/week47/html/._week47-bs025.html index b109dfad6..cc06971f2 100644 --- a/doc/pub/week47/html/._week47-bs025.html +++ b/doc/pub/week47/html/._week47-bs025.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,38 +159,40 @@ MathJax.Hub.Config({ -

    The problem to solve

    -Using our definition of the kernel We can rewrite again the Lagrangian -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, -$$ - -subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem -$$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, -$$ - -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). -If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \). +

    Different kernels and Mercer's theorem

    -We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type +There are several popular kernels being used. These are + +

      +
    1. Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),
      2. +
    2. Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),
      3. +
    3. Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),
      4. +
    4. Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),
      5. +
    + +and many other ones. + +

    +An important theorem for us is Mercer's +theorem. The +theorem states that if a kernel function \( K \) is symmetric, continuous +and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there +exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into +another space (possibly with much higher dimensions) such that + $$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ -Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \). -Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into -\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \). +

    +So you can use \( K \) as a kernel since you know \( \phi \) exists, even if +you don’t know what \( \phi \) is. + +

    +Note that some frequently used kernels (such as the Sigmoid kernel) +don’t respect all of Mercer’s conditions, yet they generally work well +in practice.

    @@ -215,7 +215,6 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.

  • 29
  • 30
  • 31
    • -
  • 32
  • »
    • diff --git a/doc/pub/week47/html/._week47-bs026.html b/doc/pub/week47/html/._week47-bs026.html index 43a5e09c6..2b1ec9580 100644 --- a/doc/pub/week47/html/._week47-bs026.html +++ b/doc/pub/week47/html/._week47-bs026.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,41 +159,199 @@ MathJax.Hub.Config({ -

    Different kernels and Mercer's theorem

    - +

    The moons example

    -There are several popular kernels being used. These are -

      -
    1. Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),
      2. -
    2. Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),
      3. -
    3. Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),
      4. -
    4. Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),
      5. -
    + +
    from __future__ import division, print_function, unicode_literals
 
-and many other ones.
+import numpy as np
+np.random.seed(42)
 
-
    
-An important theorem for us is Mercer's
-theorem.  The
-theorem states that if a kernel function \( K \) is symmetric, continuous
-and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there
-exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into
-another space (possibly with much higher dimensions) such that
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
 
-$$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
-$$
 
-
    
-So you can use \( K \) as a kernel since you know \( \phi \) exists, even if
-you don’t know what \( \phi \) is.
+from sklearn.svm import SVC
+from sklearn import datasets
 
-
    
-Note that some frequently used kernels (such as the Sigmoid kernel)
-don’t respect all of Mercer’s conditions, yet they generally work well
-in practice.
 
+
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+
+
+from sklearn.datasets import make_moons
+X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
+
+def plot_dataset(X, y, axes):
+    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
+    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
+    plt.axis(axes)
+    plt.grid(True, which='both')
+    plt.xlabel(r"$x_1$", fontsize=20)
+    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.show()
+
+from sklearn.datasets import make_moons
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+
+polynomial_svm_clf = Pipeline([
+        ("poly_features", PolynomialFeatures(degree=3)),
+        ("scaler", StandardScaler()),
+        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
+    ])
+
+polynomial_svm_clf.fit(X, y)
+
+def plot_predictions(clf, axes):
+    x0s = np.linspace(axes[0], axes[1], 100)
+    x1s = np.linspace(axes[2], axes[3], 100)
+    x0, x1 = np.meshgrid(x0s, x1s)
+    X = np.c_[x0.ravel(), x1.ravel()]
+    y_pred = clf.predict(X).reshape(x0.shape)
+    y_decision = clf.decision_function(X).reshape(x0.shape)
+    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
+    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
+
+plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+
+plt.show()
+
+
+from sklearn.svm import SVC
+
+poly_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
+    ])
+poly_kernel_svm_clf.fit(X, y)
+
+poly100_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
+    ])
+poly100_kernel_svm_clf.fit(X, y)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=3, r=1, C=5$", fontsize=18)
+
+plt.subplot(122)
+plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=10, r=100, C=5$", fontsize=18)
+
+plt.show()
+
+def gaussian_rbf(x, landmark, gamma):
+    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
+
+gamma = 0.3
+
+x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
+x2s = gaussian_rbf(x1s, -2, gamma)
+x3s = gaussian_rbf(x1s, 1, gamma)
+
+XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
+yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
+plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
+plt.plot(x1s, x2s, "g--")
+plt.plot(x1s, x3s, "b:")
+plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"Similarity", fontsize=14)
+plt.annotate(r'$\mathbf{x}$',
+             xy=(X1D[3, 0], 0),
+             xytext=(-0.5, 0.20),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
+plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
+plt.axis([-4.5, 4.5, -0.1, 1.1])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
+plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
+plt.xlabel(r"$x_2$", fontsize=20)
+plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
+plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
+             xy=(XK[3, 0], XK[3, 1]),
+             xytext=(0.65, 0.50),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
+plt.axis([-0.1, 1.1, -0.1, 1.1])
+    
+plt.subplots_adjust(right=1)
+
+plt.show()
+
+
+x1_example = X1D[3, 0]
+for landmark in (-2, 1):
+    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
+    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
+
+rbf_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
+    ])
+rbf_kernel_svm_clf.fit(X, y)
+
+
+from sklearn.svm import SVC
+
+gamma1, gamma2 = 0.1, 5
+C1, C2 = 0.001, 1000
+hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
+
+svm_clfs = []
+for gamma, C in hyperparams:
+    rbf_kernel_svm_clf = Pipeline([
+            ("scaler", StandardScaler()),
+            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
+        ])
+    rbf_kernel_svm_clf.fit(X, y)
+    svm_clfs.append(rbf_kernel_svm_clf)
+
+plt.figure(figsize=(11, 7))
+
+for i, svm_clf in enumerate(svm_clfs):
+    plt.subplot(221 + i)
+    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
+    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+    gamma, C = hyperparams[i]
+    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
+
+plt.show()
+

    @@ -216,7 +372,6 @@ in practice.

  • 29
  • 30
  • 31
    • -
  • 32
  • »
    • diff --git a/doc/pub/week47/html/._week47-bs027.html b/doc/pub/week47/html/._week47-bs027.html index 4891a99d3..3aac06e09 100644 --- a/doc/pub/week47/html/._week47-bs027.html +++ b/doc/pub/week47/html/._week47-bs027.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,199 +159,28 @@ MathJax.Hub.Config({ -

    The moons example

    +

    Mathematical optimization of convex functions

    +

    +A mathematical (quadratic) optimization problem, or just optimization problem, has the form +$$ +\begin{align*} + &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber + &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. +\end{align*} +$$ - -

    from __future__ import division, print_function, unicode_literals
+subject to some constraints for say a selected set \( i=1,2,\dots, n \).
+In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the
+vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with.
 
-import numpy as np
-np.random.seed(42)
+
    
+In our case we are particularly interested in a class of optimization problems called convex optmization problems. 
+In our discussion on gradient descent methods we discussed at length the definition of a convex function.
 
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
+
    
+Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics.
 
-
-from sklearn.svm import SVC
-from sklearn import datasets
-
-
-
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import StandardScaler
-from sklearn.svm import LinearSVC
-
-
-from sklearn.datasets import make_moons
-X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
-
-def plot_dataset(X, y, axes):
-    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
-    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
-    plt.axis(axes)
-    plt.grid(True, which='both')
-    plt.xlabel(r"$x_1$", fontsize=20)
-    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.show()
-
-from sklearn.datasets import make_moons
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import PolynomialFeatures
-
-polynomial_svm_clf = Pipeline([
-        ("poly_features", PolynomialFeatures(degree=3)),
-        ("scaler", StandardScaler()),
-        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
-    ])
-
-polynomial_svm_clf.fit(X, y)
-
-def plot_predictions(clf, axes):
-    x0s = np.linspace(axes[0], axes[1], 100)
-    x1s = np.linspace(axes[2], axes[3], 100)
-    x0, x1 = np.meshgrid(x0s, x1s)
-    X = np.c_[x0.ravel(), x1.ravel()]
-    y_pred = clf.predict(X).reshape(x0.shape)
-    y_decision = clf.decision_function(X).reshape(x0.shape)
-    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
-    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
-
-plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-
-plt.show()
-
-
-from sklearn.svm import SVC
-
-poly_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
-    ])
-poly_kernel_svm_clf.fit(X, y)
-
-poly100_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
-    ])
-poly100_kernel_svm_clf.fit(X, y)
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=3, r=1, C=5$", fontsize=18)
-
-plt.subplot(122)
-plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=10, r=100, C=5$", fontsize=18)
-
-plt.show()
-
-def gaussian_rbf(x, landmark, gamma):
-    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
-
-gamma = 0.3
-
-x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
-x2s = gaussian_rbf(x1s, -2, gamma)
-x3s = gaussian_rbf(x1s, 1, gamma)
-
-XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
-yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
-plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
-plt.plot(x1s, x2s, "g--")
-plt.plot(x1s, x3s, "b:")
-plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"Similarity", fontsize=14)
-plt.annotate(r'$\mathbf{x}$',
-             xy=(X1D[3, 0], 0),
-             xytext=(-0.5, 0.20),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
-plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
-plt.axis([-4.5, 4.5, -0.1, 1.1])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
-plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
-plt.xlabel(r"$x_2$", fontsize=20)
-plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
-plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
-             xy=(XK[3, 0], XK[3, 1]),
-             xytext=(0.65, 0.50),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
-plt.axis([-0.1, 1.1, -0.1, 1.1])
-    
-plt.subplots_adjust(right=1)
-
-plt.show()
-
-
-x1_example = X1D[3, 0]
-for landmark in (-2, 1):
-    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
-    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
-
-rbf_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
-    ])
-rbf_kernel_svm_clf.fit(X, y)
-
-
-from sklearn.svm import SVC
-
-gamma1, gamma2 = 0.1, 5
-C1, C2 = 0.001, 1000
-hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
-
-svm_clfs = []
-for gamma, C in hyperparams:
-    rbf_kernel_svm_clf = Pipeline([
-            ("scaler", StandardScaler()),
-            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
-        ])
-    rbf_kernel_svm_clf.fit(X, y)
-    svm_clfs.append(rbf_kernel_svm_clf)
-
-plt.figure(figsize=(11, 7))
-
-for i, svm_clf in enumerate(svm_clfs):
-    plt.subplot(221 + i)
-    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
-    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-    gamma, C = hyperparams[i]
-    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
-
-plt.show()
-

    @@ -373,7 +200,6 @@ plt.show()

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    • diff --git a/doc/pub/week47/html/._week47-bs028.html b/doc/pub/week47/html/._week47-bs028.html index da0f5666a..ca5315deb 100644 --- a/doc/pub/week47/html/._week47-bs028.html +++ b/doc/pub/week47/html/._week47-bs028.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,27 +159,28 @@ MathJax.Hub.Config({ -

    Mathematical optimization of convex functions

    +

    How do we solve these problems?

    -A mathematical (quadratic) optimization problem, or just optimization problem, has the form -$$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -$$ - -subject to some constraints for say a selected set \( i=1,2,\dots, n \). -In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the -vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with. +If we use Python as programming language and wish to venture beyond +scikit-learn, tensorflow and similar software which makes our +lives so much easier, we need to dive into the wonderful world of +quadratic programming. We can, if we wish, solve the minimization +problem using say standard gradient methods or conjugate gradient +methods. However, these methods tend to exhibit a rather slow +converge. So, welcome to the promised land of quadratic programming.

    -In our case we are particularly interested in a class of optimization problems called convex optmization problems. -In our discussion on gradient descent methods we discussed at length the definition of a convex function. +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as

    -Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics. + + +

    import numpy
+import cvxopt
+
    +

    +This will make our life much easier. You don't need t write your own optimizer.

    @@ -201,7 +200,6 @@ Convex optimization problems play a central role in applied mathematics and we r

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    • diff --git a/doc/pub/week47/html/._week47-bs029.html b/doc/pub/week47/html/._week47-bs029.html index ca9d4386f..d58fbe6e1 100644 --- a/doc/pub/week47/html/._week47-bs029.html +++ b/doc/pub/week47/html/._week47-bs029.html @@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source ('Getting into the details', 2, None, '___sec9'), ('First attempt at a minimization approach', 2, None, '___sec10'), ('Solving the equations', 2, None, '___sec11'), - ('Code Example', 2, None, '___sec12'), - ('Problems with the Simpler Approach', 2, None, '___sec13'), - ('A better approach', 2, None, '___sec14'), + ('Can we code this?', 2, None, '___sec12'), + ('A better approach', 2, None, '___sec13'), ('A quick Reminder on Lagrangian Multipliers', 2, None, - '___sec15'), - ('Adding the Multiplier', 2, None, '___sec16'), - ('Setting up the Problem', 2, None, '___sec17'), - ('The problem to solve', 2, None, '___sec18'), - ('The last steps', 2, None, '___sec19'), - ('A soft classifier', 2, None, '___sec20'), - ('Soft optmization problem', 2, None, '___sec21'), - ('Kernels and non-linearity', 2, None, '___sec22'), - ('The equations', 2, None, '___sec23'), - ('The problem to solve', 2, None, '___sec24'), - ("Different kernels and Mercer's theorem", 2, None, '___sec25'), - ('The moons example', 2, None, '___sec26'), + '___sec14'), + ('Adding the Multiplier', 2, None, '___sec15'), + ('Setting up the Problem', 2, None, '___sec16'), + ('The problem to solve', 2, None, '___sec17'), + ('The last steps', 2, None, '___sec18'), + ('A soft classifier', 2, None, '___sec19'), + ('Soft optmization problem', 2, None, '___sec20'), + ('Kernels and non-linearity', 2, None, '___sec21'), + ('The equations', 2, None, '___sec22'), + ('The problem to solve', 2, None, '___sec23'), + ("Different kernels and Mercer's theorem", 2, None, '___sec24'), + ('The moons example', 2, None, '___sec25'), ('Mathematical optimization of convex functions', 2, None, - '___sec27'), - ('How do we solve these problems?', 2, None, '___sec28'), - ('A simple example', 2, None, '___sec29'), - ('Back to the more realistic cases', 2, None, '___sec30')]} + '___sec26'), + ('How do we solve these problems?', 2, None, '___sec27'), + ('A simple example', 2, None, '___sec28'), + ('Back to the more realistic cases', 2, None, '___sec29')]} end of tocinfo --> @@ -127,25 +126,24 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • +
  • Can we code this?
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • @@ -161,29 +159,71 @@ MathJax.Hub.Config({ -

    How do we solve these problems?

    +

    A simple example

    -If we use Python as programming language and wish to venture beyond -scikit-learn, tensorflow and similar software which makes our -lives so much easier, we need to dive into the wonderful world of -quadratic programming. We can, if we wish, solve the minimization -problem using say standard gradient methods or conjugate gradient -methods. However, these methods tend to exhibit a rather slow -converge. So, welcome to the promised land of quadratic programming. +We remind ourselves about the general problem we want to solve +$$ +\begin{align*} + &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber + &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. +\end{align*} +$$

    -The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as +Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem +$$ +\begin{align*} + &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber + &\mathrm{subject to} \\ \nonumber + &x, y \geq 0 \\ \nonumber + &x+3y \geq 15 \\ \nonumber + &2x+5y \leq 100 \\ \nonumber + &3x+4y \leq 80. \\ \nonumber +\end{align*} +$$ +The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns) +$$ +\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. +$$ + +Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation +$$ +\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. +$$ + +We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix +$$ +\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} +$$ + +is clearly positive semi-definite (all eigenvalues larger or equal zero). +Finally, the vector \( \boldsymbol{h} \) is defined as +$$ +\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. +$$ + +

    +Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero +The following code solves the equations for us

    -

    import numpy
-import cvxopt
+
    # Import the necessary packages
+import numpy
+from cvxopt import matrix
+from cvxopt import solvers
+P = matrix(numpy.diag([1,0]), tc=’d’)
+q = matrix(numpy.array([3,4]), tc=’d’)
+G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
+h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
+# Construct the QP, invoke solver
+sol = solvers.qp(P,q,G,h)
+# Extract optimal value and solution
+sol[’x’] 
+sol[’primal objective’]
 
    
-
    
-This will make our life much easier. You don't need t write your own optimizer.
-
 
    
 
    
 
@@ -201,7 +241,6 @@ This will make our life much easier. You don't need t write your own optimizer.
   
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    • 
 
 
diff --git a/doc/pub/week47/html/._week47-bs030.html b/doc/pub/week47/html/._week47-bs030.html
index 1a7eb37b6..796dc2e6f 100644
--- a/doc/pub/week47/html/._week47-bs030.html
+++ b/doc/pub/week47/html/._week47-bs030.html
@@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source
               ('Getting into the details', 2, None, '___sec9'),
               ('First attempt at a minimization approach', 2, None, '___sec10'),
               ('Solving the equations', 2, None, '___sec11'),
-              ('Code Example', 2, None, '___sec12'),
-              ('Problems with the Simpler Approach', 2, None, '___sec13'),
-              ('A better approach', 2, None, '___sec14'),
+              ('Can we code this?', 2, None, '___sec12'),
+              ('A better approach', 2, None, '___sec13'),
               ('A quick Reminder on Lagrangian Multipliers',
                2,
                None,
-               '___sec15'),
-              ('Adding the Multiplier', 2, None, '___sec16'),
-              ('Setting up the Problem', 2, None, '___sec17'),
-              ('The problem to solve', 2, None, '___sec18'),
-              ('The last steps', 2, None, '___sec19'),
-              ('A soft classifier', 2, None, '___sec20'),
-              ('Soft optmization problem', 2, None, '___sec21'),
-              ('Kernels and non-linearity', 2, None, '___sec22'),
-              ('The equations', 2, None, '___sec23'),
-              ('The problem to solve', 2, None, '___sec24'),
-              ("Different kernels and Mercer's theorem", 2, None, '___sec25'),
-              ('The moons example', 2, None, '___sec26'),
+               '___sec14'),
+              ('Adding the Multiplier', 2, None, '___sec15'),
+              ('Setting up the Problem', 2, None, '___sec16'),
+              ('The problem to solve', 2, None, '___sec17'),
+              ('The last steps', 2, None, '___sec18'),
+              ('A soft classifier', 2, None, '___sec19'),
+              ('Soft optmization problem', 2, None, '___sec20'),
+              ('Kernels and non-linearity', 2, None, '___sec21'),
+              ('The equations', 2, None, '___sec22'),
+              ('The problem to solve', 2, None, '___sec23'),
+              ("Different kernels and Mercer's theorem", 2, None, '___sec24'),
+              ('The moons example', 2, None, '___sec25'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec27'),
-              ('How do we solve these problems?', 2, None, '___sec28'),
-              ('A simple example', 2, None, '___sec29'),
-              ('Back to the more realistic cases', 2, None, '___sec30')]}
+               '___sec26'),
+              ('How do we solve these problems?', 2, None, '___sec27'),
+              ('A simple example', 2, None, '___sec28'),
+              ('Back to the more realistic cases', 2, None, '___sec29')]}
 end of tocinfo -->
 
 
@@ -127,25 +126,24 @@ MathJax.Hub.Config({
       
  • Getting into the details
    • 
       
  • First attempt at a minimization approach
    • 
       
  • Solving the equations
    • 
-      
  • Code Example
    • 
-      
  • Problems with the Simpler Approach
    • 
-      
  • A better approach
    • 
-      
  • A quick Reminder on Lagrangian Multipliers
    • 
-      
  • Adding the Multiplier
    • 
-      
  • Setting up the Problem
    • 
-      
  • The problem to solve
    • 
-      
  • The last steps
    • 
-      
  • A soft classifier
    • 
-      
  • Soft optmization problem
    • 
-      
  • Kernels and non-linearity
    • 
-      
  • The equations
    • 
-      
  • The problem to solve
    • 
-      
  • Different kernels and Mercer's theorem
    • 
-      
  • The moons example
    • 
-      
  • Mathematical optimization of convex functions
    • 
-      
  • How do we solve these problems?
    • 
-      
  • A simple example
    • 
-      
  • Back to the more realistic cases
    • 
+      
  • Can we code this?
    • 
+      
  • A better approach
    • 
+      
  • A quick Reminder on Lagrangian Multipliers
    • 
+      
  • Adding the Multiplier
    • 
+      
  • Setting up the Problem
    • 
+      
  • The problem to solve
    • 
+      
  • The last steps
    • 
+      
  • A soft classifier
    • 
+      
  • Soft optmization problem
    • 
+      
  • Kernels and non-linearity
    • 
+      
  • The equations
    • 
+      
  • The problem to solve
    • 
+      
  • Different kernels and Mercer's theorem
    • 
+      
  • The moons example
    • 
+      
  • Mathematical optimization of convex functions
    • 
+      
  • How do we solve these problems?
    • 
+      
  • A simple example
    • 
+      
  • Back to the more realistic cases
    • 
 
         
       
@@ -161,72 +159,25 @@ MathJax.Hub.Config({
 
 
 
-
    A simple example 
    
+
    Back to the more realistic cases 
    
 
 
    
-We remind ourselves about the general problem we want to solve
+We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have
 $$
-\begin{align*}
-    &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber
-    &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge  \boldsymbol{A}\boldsymbol{x}=f.
-\end{align*}
+\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\
+y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\
+\dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots \\
+y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\
+\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, 
 $$
 
+subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and 
+\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). 
+With  the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).
+
 
    
-Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem
-$$
-\begin{align*}
-    &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber
-    &\mathrm{subject to} \\ \nonumber
-    &x, y \geq 0 \\ \nonumber
-    &x+3y  \geq 15 \\ \nonumber
-    &2x+5y  \leq  100 \\ \nonumber
-    &3x+4y  \leq  80.  \\ \nonumber
-\end{align*}
-$$
 
-The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns)
-$$
-\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T   \begin{bmatrix} 1 & 0\\ 0 & 0  \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  + \begin{bmatrix}3\\ 4  \end{bmatrix}^T \begin{bmatrix}x \\ y  \end{bmatrix}.  
-$$
-
-Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation
-$$
-\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
-$$
-
-We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix 
-$$
-\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0  \end{bmatrix}
-$$
-
-is clearly positive semi-definite (all eigenvalues larger or equal zero). 
-Finally, the vector \( \boldsymbol{h} \) is defined as 
-$$
-\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
-$$
-
-
    
-Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero
-The following code solves the equations for us
-
    
-
-
-
    # Import the necessary packages
-import numpy
-from cvxopt import matrix
-from cvxopt import solvers
-P = matrix(numpy.diag([1,0]), tc=’d’)
-q = matrix(numpy.array([3,4]), tc=’d’)
-G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
-h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
-# Construct the QP, invoke solver
-sol = solvers.qp(P,q,G,h)
-# Extract optimal value and solution
-sol[’x’] 
-sol[’primal objective’]
-
    
-
    
 
    
 
 
      
@@ -242,8 +193,6 @@ sol[’primal objective’]
   
    • 29
      • 
   
    • 30
      • 
   
    • 31
      • 
-  
    • 32
      • 
-  
    • »
      • 
 
    
 
 
diff --git a/doc/pub/week47/html/week47-bs.html b/doc/pub/week47/html/week47-bs.html
index e96c3a089..f6f7353f5 100644
--- a/doc/pub/week47/html/week47-bs.html
+++ b/doc/pub/week47/html/week47-bs.html
@@ -53,31 +53,30 @@ Automatically generated HTML file from DocOnce source
               ('Getting into the details', 2, None, '___sec9'),
               ('First attempt at a minimization approach', 2, None, '___sec10'),
               ('Solving the equations', 2, None, '___sec11'),
-              ('Code Example', 2, None, '___sec12'),
-              ('Problems with the Simpler Approach', 2, None, '___sec13'),
-              ('A better approach', 2, None, '___sec14'),
+              ('Can we code this?', 2, None, '___sec12'),
+              ('A better approach', 2, None, '___sec13'),
               ('A quick Reminder on Lagrangian Multipliers',
                2,
                None,
-               '___sec15'),
-              ('Adding the Multiplier', 2, None, '___sec16'),
-              ('Setting up the Problem', 2, None, '___sec17'),
-              ('The problem to solve', 2, None, '___sec18'),
-              ('The last steps', 2, None, '___sec19'),
-              ('A soft classifier', 2, None, '___sec20'),
-              ('Soft optmization problem', 2, None, '___sec21'),
-              ('Kernels and non-linearity', 2, None, '___sec22'),
-              ('The equations', 2, None, '___sec23'),
-              ('The problem to solve', 2, None, '___sec24'),
-              ("Different kernels and Mercer's theorem", 2, None, '___sec25'),
-              ('The moons example', 2, None, '___sec26'),
+               '___sec14'),
+              ('Adding the Multiplier', 2, None, '___sec15'),
+              ('Setting up the Problem', 2, None, '___sec16'),
+              ('The problem to solve', 2, None, '___sec17'),
+              ('The last steps', 2, None, '___sec18'),
+              ('A soft classifier', 2, None, '___sec19'),
+              ('Soft optmization problem', 2, None, '___sec20'),
+              ('Kernels and non-linearity', 2, None, '___sec21'),
+              ('The equations', 2, None, '___sec22'),
+              ('The problem to solve', 2, None, '___sec23'),
+              ("Different kernels and Mercer's theorem", 2, None, '___sec24'),
+              ('The moons example', 2, None, '___sec25'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec27'),
-              ('How do we solve these problems?', 2, None, '___sec28'),
-              ('A simple example', 2, None, '___sec29'),
-              ('Back to the more realistic cases', 2, None, '___sec30')]}
+               '___sec26'),
+              ('How do we solve these problems?', 2, None, '___sec27'),
+              ('A simple example', 2, None, '___sec28'),
+              ('Back to the more realistic cases', 2, None, '___sec29')]}
 end of tocinfo -->
 
 
@@ -127,25 +126,24 @@ MathJax.Hub.Config({
       
  • Getting into the details
    • 
       
  • First attempt at a minimization approach
    • 
       
  • Solving the equations
    • 
-      
  • Code Example
    • 
-      
  • Problems with the Simpler Approach
    • 
-      
  • A better approach
    • 
-      
  • A quick Reminder on Lagrangian Multipliers
    • 
-      
  • Adding the Multiplier
    • 
-      
  • Setting up the Problem
    • 
-      
  • The problem to solve
    • 
-      
  • The last steps
    • 
-      
  • A soft classifier
    • 
-      
  • Soft optmization problem
    • 
-      
  • Kernels and non-linearity
    • 
-      
  • The equations
    • 
-      
  • The problem to solve
    • 
-      
  • Different kernels and Mercer's theorem
    • 
-      
  • The moons example
    • 
-      
  • Mathematical optimization of convex functions
    • 
-      
  • How do we solve these problems?
    • 
-      
  • A simple example
    • 
-      
  • Back to the more realistic cases
    • 
+      
  • Can we code this?
    • 
+      
  • A better approach
    • 
+      
  • A quick Reminder on Lagrangian Multipliers
    • 
+      
  • Adding the Multiplier
    • 
+      
  • Setting up the Problem
    • 
+      
  • The problem to solve
    • 
+      
  • The last steps
    • 
+      
  • A soft classifier
    • 
+      
  • Soft optmization problem
    • 
+      
  • Kernels and non-linearity
    • 
+      
  • The equations
    • 
+      
  • The problem to solve
    • 
+      
  • Different kernels and Mercer's theorem
    • 
+      
  • The moons example
    • 
+      
  • Mathematical optimization of convex functions
    • 
+      
  • How do we solve these problems?
    • 
+      
  • A simple example
    • 
+      
  • Back to the more realistic cases
    • 
 
         
       
@@ -204,7 +202,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 32
    • 
+  
  • 31
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week47/html/week47-reveal.html b/doc/pub/week47/html/week47-reveal.html
index b29673c88..992d61b64 100644
--- a/doc/pub/week47/html/week47-reveal.html
+++ b/doc/pub/week47/html/week47-reveal.html
@@ -524,26 +524,19 @@ where \( \eta \) is our by now well-known learning rate.
 
 
 
    
-
    Code Example 
    
+
    Can we code this?  
    
 
 
    
 The equations we discussed above can be coded rather easily (the
-framework is similar to what we developed for logistic
-regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.
-
    
-
-
-
    
-
    
-
    
-
-
-
    
-
    Problems with the Simpler Approach 
    
+framework is similar to what we developed for logistic regression). We
+can set up a simple case with two classes only and we want to find a
+line which separates them the best possible way.
 
 
    
 There are however problems with this approach, although it looks
-pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+pretty straightforward to implement. When running a code for such a
+case we can easily end up with many diffeent lines which separate the
+two classes.
 
 
    
 For small
@@ -555,7 +548,7 @@ at all.
 
 
 
    
-
    A better approach 
    
+
    A better approach 
    
 
 
    
 A better approach is rather to try to define a large margin between
@@ -605,7 +598,7 @@ about Lagrangian multipliers.
 
 
 
    
-
    A quick Reminder on Lagrangian Multipliers 
    
+
    A quick Reminder on Lagrangian Multipliers 
    
 
 
    
 Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
@@ -667,7 +660,7 @@ Then \( dz \) is no longer arbitrary.
 
 
 
    
-
    Adding the Multiplier 
    
+
    Adding the Multiplier 
    
 
 
    
 However, we can add to
@@ -722,7 +715,7 @@ $$
 
 
 
    
-
    Setting up the Problem 
    
+
    Setting up the Problem 
    
 In order to solve the above problem, we define the following Lagrangian function to be minimized 
 
     
    
 $$
@@ -774,7 +767,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
 
 
 
    
-
    The problem to solve 
    
+
    The problem to solve 
    
 
 
    
 We can rewrite 
@@ -802,7 +795,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
 
 
 
    
-
    The last steps 
    
+
    The last steps 
    
 
 
    
 Solving the above problem, yields the values of \( \lambda_i \).
@@ -846,7 +839,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
 
 
 
    
-
    A soft classifier 
    
+
    A soft classifier 
    
 
 
    
 Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
@@ -885,7 +878,7 @@ misclassifications.
 
 
 
    
-
    Soft optmization problem 
    
+
    Soft optmization problem 
    
 
 
    
 This has in turn the consequences that we change our optmization problem to finding the minimum of 
@@ -957,7 +950,7 @@ $$
 
 
 
    
-
    Kernels and non-linearity 
    
+
    Kernels and non-linearity 
    
 
 
    
 The cases we have studied till now, were all characterized by two classes
@@ -1031,7 +1024,7 @@ plt.show()
 
 
 
    
-
    The equations 
    
+
    The equations 
    
 
 
    
 Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
@@ -1086,7 +1079,7 @@ the trouble of performing the transformation
 
 
 
    
-
    The problem to solve 
    
+
    The problem to solve 
    
 Using our definition of the kernel We can rewrite again the Lagrangian
 
     
    
 $$
@@ -1128,7 +1121,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
 
 
 
    
-
    Different kernels and Mercer's theorem 
    
+
    Different kernels and Mercer's theorem 
    
 
 
    
 There are several popular kernels being used. These are
@@ -1169,7 +1162,7 @@ in practice.
 
 
 
    
-
    The moons example 
    
+
    The moons example 
    
 
    
 
 
@@ -1366,7 +1359,7 @@ plt.show()
 
 
 
    
-
    Mathematical optimization of convex functions 
    
+
    Mathematical optimization of convex functions 
    
 
 
    
 A mathematical (quadratic) optimization problem, or just optimization problem, has the form
@@ -1393,7 +1386,7 @@ Convex optimization problems play a central role in applied mathematics and we r
 
 
 
    
-
    How do we solve these problems? 
    
+
    How do we solve these problems? 
    
 
 
    
 If we use Python as programming language and wish to venture beyond
@@ -1419,7 +1412,7 @@ This will make our life much easier. You don't need t write your own optimizer.
 
 
 
    
-
    A simple example 
    
+
    A simple example 
    
 
 
    
 We remind ourselves about the general problem we want to solve
@@ -1500,7 +1493,7 @@ sol[’primal obj
 
 
 
    
-
    Back to the more realistic cases 
    
+
    Back to the more realistic cases 
    
 
 
    
 We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have
diff --git a/doc/pub/week47/html/week47-solarized.html b/doc/pub/week47/html/week47-solarized.html
index d63bf6582..c22722275 100644
--- a/doc/pub/week47/html/week47-solarized.html
+++ b/doc/pub/week47/html/week47-solarized.html
@@ -47,31 +47,30 @@ div { text-align: justify; text-justify: inter-word; }
               ('Getting into the details', 2, None, '___sec9'),
               ('First attempt at a minimization approach', 2, None, '___sec10'),
               ('Solving the equations', 2, None, '___sec11'),
-              ('Code Example', 2, None, '___sec12'),
-              ('Problems with the Simpler Approach', 2, None, '___sec13'),
-              ('A better approach', 2, None, '___sec14'),
+              ('Can we code this?', 2, None, '___sec12'),
+              ('A better approach', 2, None, '___sec13'),
               ('A quick Reminder on Lagrangian Multipliers',
                2,
                None,
-               '___sec15'),
-              ('Adding the Multiplier', 2, None, '___sec16'),
-              ('Setting up the Problem', 2, None, '___sec17'),
-              ('The problem to solve', 2, None, '___sec18'),
-              ('The last steps', 2, None, '___sec19'),
-              ('A soft classifier', 2, None, '___sec20'),
-              ('Soft optmization problem', 2, None, '___sec21'),
-              ('Kernels and non-linearity', 2, None, '___sec22'),
-              ('The equations', 2, None, '___sec23'),
-              ('The problem to solve', 2, None, '___sec24'),
-              ("Different kernels and Mercer's theorem", 2, None, '___sec25'),
-              ('The moons example', 2, None, '___sec26'),
+               '___sec14'),
+              ('Adding the Multiplier', 2, None, '___sec15'),
+              ('Setting up the Problem', 2, None, '___sec16'),
+              ('The problem to solve', 2, None, '___sec17'),
+              ('The last steps', 2, None, '___sec18'),
+              ('A soft classifier', 2, None, '___sec19'),
+              ('Soft optmization problem', 2, None, '___sec20'),
+              ('Kernels and non-linearity', 2, None, '___sec21'),
+              ('The equations', 2, None, '___sec22'),
+              ('The problem to solve', 2, None, '___sec23'),
+              ("Different kernels and Mercer's theorem", 2, None, '___sec24'),
+              ('The moons example', 2, None, '___sec25'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec27'),
-              ('How do we solve these problems?', 2, None, '___sec28'),
-              ('A simple example', 2, None, '___sec29'),
-              ('Back to the more realistic cases', 2, None, '___sec30')]}
+               '___sec26'),
+              ('How do we solve these problems?', 2, None, '___sec27'),
+              ('A simple example', 2, None, '___sec28'),
+              ('Back to the more realistic cases', 2, None, '___sec29')]}
 end of tocinfo -->
 
 
@@ -452,25 +451,19 @@ where \( \eta \) is our by now well-known learning rate.
 
    
 









    
 
-
    Code Example 
    
+
    Can we code this?  
    
 
 
    
 The equations we discussed above can be coded rather easily (the
-framework is similar to what we developed for logistic
-regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.
-
    
-
-
-
    
-
    
-
    
-









    
-
-
    Problems with the Simpler Approach 
    
+framework is similar to what we developed for logistic regression). We
+can set up a simple case with two classes only and we want to find a
+line which separates them the best possible way.
 
 
    
 There are however problems with this approach, although it looks
-pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+pretty straightforward to implement. When running a code for such a
+case we can easily end up with many diffeent lines which separate the
+two classes.
 
 
    
 For small
@@ -482,7 +475,7 @@ at all.
 
    
 









    
 
-
    A better approach 
    
+
    A better approach 
    
 
 
    
 A better approach is rather to try to define a large margin between
@@ -524,7 +517,7 @@ about Lagrangian multipliers.
 
    
 









    
 
-
    A quick Reminder on Lagrangian Multipliers 
    
+
    A quick Reminder on Lagrangian Multipliers 
    
 
 
    
 Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
@@ -574,7 +567,7 @@ Then \( dz \) is no longer arbitrary.
 
    
 









    
 
-
    Adding the Multiplier 
    
+
    Adding the Multiplier 
    
 
 
    
 However, we can add to
@@ -617,7 +610,7 @@ $$
 
    
 









    
 
-
    Setting up the Problem 
    
+
    Setting up the Problem 
    
 In order to solve the above problem, we define the following Lagrangian function to be minimized 
 $$
 {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
@@ -658,7 +651,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 
 
    
 We can rewrite 
@@ -682,7 +675,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
 
    
 









    
 
-
    The last steps 
    
+
    The last steps 
    
 
 
    
 Solving the above problem, yields the values of \( \lambda_i \).
@@ -716,7 +709,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
 
    
 









    
 
-
    A soft classifier 
    
+
    A soft classifier 
    
 
 
    
 Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
@@ -751,7 +744,7 @@ misclassifications.
 
    
 









    
 
-
    Soft optmization problem 
    
+
    Soft optmization problem 
    
 
 
    
 This has in turn the consequences that we change our optmization problem to finding the minimum of 
@@ -805,7 +798,7 @@ $$
 
    
 









    
 
-
    Kernels and non-linearity 
    
+
    Kernels and non-linearity 
    
 
 
    
 The cases we have studied till now, were all characterized by two classes
@@ -878,7 +871,7 @@ plt.show()
 
    
 









    
 
-
    The equations 
    
+
    The equations 
    
 
 
    
 Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
@@ -923,7 +916,7 @@ the trouble of performing the transformation
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 Using our definition of the kernel We can rewrite again the Lagrangian
 $$
 {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
@@ -959,7 +952,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
 
    
 









    
 
-
    Different kernels and Mercer's theorem 
    
+
    Different kernels and Mercer's theorem 
    
 
 
    
 There are several popular kernels being used. These are
@@ -997,7 +990,7 @@ in practice.
 
    
 









    
 
-
    The moons example 
    
+
    The moons example 
    
 
    
 
 
@@ -1193,7 +1186,7 @@ plt.show()
 
    
 









    
 
-
    Mathematical optimization of convex functions 
    
+
    Mathematical optimization of convex functions 
    
 
 
    
 A mathematical (quadratic) optimization problem, or just optimization problem, has the form
@@ -1218,7 +1211,7 @@ Convex optimization problems play a central role in applied mathematics and we r
 
    
 









    
 
-
    How do we solve these problems? 
    
+
    How do we solve these problems? 
    
 
 
    
 If we use Python as programming language and wish to venture beyond
@@ -1244,7 +1237,7 @@ This will make our life much easier. You don't need t write your own optimizer.
 
    
 









    
 
-
    A simple example 
    
+
    A simple example 
    
 
 
    
 We remind ourselves about the general problem we want to solve
@@ -1312,7 +1305,7 @@ sol[’primal obj
 
    
 









    
 
-
    Back to the more realistic cases 
    
+
    Back to the more realistic cases 
    
 
 
    
 We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have
diff --git a/doc/pub/week47/html/week47.html b/doc/pub/week47/html/week47.html
index 922be221c..5a755bc43 100644
--- a/doc/pub/week47/html/week47.html
+++ b/doc/pub/week47/html/week47.html
@@ -52,31 +52,30 @@ div { text-align: justify; text-justify: inter-word; }
               ('Getting into the details', 2, None, '___sec9'),
               ('First attempt at a minimization approach', 2, None, '___sec10'),
               ('Solving the equations', 2, None, '___sec11'),
-              ('Code Example', 2, None, '___sec12'),
-              ('Problems with the Simpler Approach', 2, None, '___sec13'),
-              ('A better approach', 2, None, '___sec14'),
+              ('Can we code this?', 2, None, '___sec12'),
+              ('A better approach', 2, None, '___sec13'),
               ('A quick Reminder on Lagrangian Multipliers',
                2,
                None,
-               '___sec15'),
-              ('Adding the Multiplier', 2, None, '___sec16'),
-              ('Setting up the Problem', 2, None, '___sec17'),
-              ('The problem to solve', 2, None, '___sec18'),
-              ('The last steps', 2, None, '___sec19'),
-              ('A soft classifier', 2, None, '___sec20'),
-              ('Soft optmization problem', 2, None, '___sec21'),
-              ('Kernels and non-linearity', 2, None, '___sec22'),
-              ('The equations', 2, None, '___sec23'),
-              ('The problem to solve', 2, None, '___sec24'),
-              ("Different kernels and Mercer's theorem", 2, None, '___sec25'),
-              ('The moons example', 2, None, '___sec26'),
+               '___sec14'),
+              ('Adding the Multiplier', 2, None, '___sec15'),
+              ('Setting up the Problem', 2, None, '___sec16'),
+              ('The problem to solve', 2, None, '___sec17'),
+              ('The last steps', 2, None, '___sec18'),
+              ('A soft classifier', 2, None, '___sec19'),
+              ('Soft optmization problem', 2, None, '___sec20'),
+              ('Kernels and non-linearity', 2, None, '___sec21'),
+              ('The equations', 2, None, '___sec22'),
+              ('The problem to solve', 2, None, '___sec23'),
+              ("Different kernels and Mercer's theorem", 2, None, '___sec24'),
+              ('The moons example', 2, None, '___sec25'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec27'),
-              ('How do we solve these problems?', 2, None, '___sec28'),
-              ('A simple example', 2, None, '___sec29'),
-              ('Back to the more realistic cases', 2, None, '___sec30')]}
+               '___sec26'),
+              ('How do we solve these problems?', 2, None, '___sec27'),
+              ('A simple example', 2, None, '___sec28'),
+              ('Back to the more realistic cases', 2, None, '___sec29')]}
 end of tocinfo -->
 
 
@@ -457,25 +456,19 @@ where \( \eta \) is our by now well-known learning rate.
 
    
 









    
 
-
    Code Example 
    
+
    Can we code this?  
    
 
 
    
 The equations we discussed above can be coded rather easily (the
-framework is similar to what we developed for logistic
-regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.
-
    
-
-
-
    
-
    
-
    
-









    
-
-
    Problems with the Simpler Approach 
    
+framework is similar to what we developed for logistic regression). We
+can set up a simple case with two classes only and we want to find a
+line which separates them the best possible way.
 
 
    
 There are however problems with this approach, although it looks
-pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+pretty straightforward to implement. When running a code for such a
+case we can easily end up with many diffeent lines which separate the
+two classes.
 
 
    
 For small
@@ -487,7 +480,7 @@ at all.
 
    
 









    
 
-
    A better approach 
    
+
    A better approach 
    
 
 
    
 A better approach is rather to try to define a large margin between
@@ -529,7 +522,7 @@ about Lagrangian multipliers.
 
    
 









    
 
-
    A quick Reminder on Lagrangian Multipliers 
    
+
    A quick Reminder on Lagrangian Multipliers 
    
 
 
    
 Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
@@ -579,7 +572,7 @@ Then \( dz \) is no longer arbitrary.
 
    
 









    
 
-
    Adding the Multiplier 
    
+
    Adding the Multiplier 
    
 
 
    
 However, we can add to
@@ -622,7 +615,7 @@ $$
 
    
 









    
 
-
    Setting up the Problem 
    
+
    Setting up the Problem 
    
 In order to solve the above problem, we define the following Lagrangian function to be minimized 
 $$
 {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
@@ -663,7 +656,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 
 
    
 We can rewrite 
@@ -687,7 +680,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
 
    
 









    
 
-
    The last steps 
    
+
    The last steps 
    
 
 
    
 Solving the above problem, yields the values of \( \lambda_i \).
@@ -721,7 +714,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
 
    
 









    
 
-
    A soft classifier 
    
+
    A soft classifier 
    
 
 
    
 Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
@@ -756,7 +749,7 @@ misclassifications.
 
    
 









    
 
-
    Soft optmization problem 
    
+
    Soft optmization problem 
    
 
 
    
 This has in turn the consequences that we change our optmization problem to finding the minimum of 
@@ -810,7 +803,7 @@ $$
 
    
 









    
 
-
    Kernels and non-linearity 
    
+
    Kernels and non-linearity 
    
 
 
    
 The cases we have studied till now, were all characterized by two classes
@@ -883,7 +876,7 @@ plt.show()
 
    
 









    
 
-
    The equations 
    
+
    The equations 
    
 
 
    
 Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
@@ -928,7 +921,7 @@ the trouble of performing the transformation
 
    
 









    
 
-
    The problem to solve 
    
+
    The problem to solve 
    
 Using our definition of the kernel We can rewrite again the Lagrangian
 $$
 {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
@@ -964,7 +957,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
 
    
 









    
 
-
    Different kernels and Mercer's theorem 
    
+
    Different kernels and Mercer's theorem 
    
 
 
    
 There are several popular kernels being used. These are
@@ -1002,7 +995,7 @@ in practice.
 
    
 









    
 
-
    The moons example 
    
+
    The moons example 
    
 
    
 
 
@@ -1198,7 +1191,7 @@ plt.show()
 
    
 









    
 
-
    Mathematical optimization of convex functions 
    
+
    Mathematical optimization of convex functions 
    
 
 
    
 A mathematical (quadratic) optimization problem, or just optimization problem, has the form
@@ -1223,7 +1216,7 @@ Convex optimization problems play a central role in applied mathematics and we r
 
    
 









    
 
-
    How do we solve these problems? 
    
+
    How do we solve these problems? 
    
 
 
    
 If we use Python as programming language and wish to venture beyond
@@ -1249,7 +1242,7 @@ This will make our life much easier. You don't need t write your own optimizer.
 
    
 









    
 
-
    A simple example 
    
+
    A simple example 
    
 
 
    
 We remind ourselves about the general problem we want to solve
@@ -1317,7 +1310,7 @@ sol[’primal objective’]
 
    
 









    
 
-
    Back to the more realistic cases 
    
+
    Back to the more realistic cases 
    
 
 
    
 We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have
diff --git a/doc/pub/week47/ipynb/ipynb-week47-src.tar.gz b/doc/pub/week47/ipynb/ipynb-week47-src.tar.gz
index 1dc28a641f4a4dde144c10efc5d873ce0ddbf9e5..4cc101cd838a744c715a85ae8c694bcae409ac9e 100644
GIT binary patch
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diff --git a/doc/pub/week47/ipynb/week47.ipynb b/doc/pub/week47/ipynb/week47.ipynb
index 7760a5e3c..e1dcc93ce 100644
--- a/doc/pub/week47/ipynb/week47.ipynb
+++ b/doc/pub/week47/ipynb/week47.ipynb
@@ -463,22 +463,18 @@
     "where $\\eta$ is our by now well-known learning rate. \n",
     "\n",
     "\n",
-    "## Code Example\n",
+    "## Can we code this?\n",
     "\n",
     "The equations we discussed above can be coded rather easily (the\n",
-    "framework is similar to what we developed for logistic\n",
-    "regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Problems with the Simpler Approach\n",
+    "framework is similar to what we developed for logistic regression). We\n",
+    "can set up a simple case with two classes only and we want to find a\n",
+    "line which separates them the best possible way.\n",
     "\n",
     "\n",
     "There are however problems with this approach, although it looks\n",
-    "pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.\n",
+    "pretty straightforward to implement. When running a code for such a\n",
+    "case we can easily end up with many diffeent lines which separate the\n",
+    "two classes.\n",
     "\n",
     "\n",
     "For small\n",
diff --git a/doc/src/week47/week47.do.txt b/doc/src/week47/week47.do.txt
index af70b14bd..76026cf62 100644
--- a/doc/src/week47/week47.do.txt
+++ b/doc/src/week47/week47.do.txt
@@ -313,21 +313,18 @@ where $\eta$ is our by now well-known learning rate.
 
 
 !split
-===== Code Example =====
+===== Can we code this?  =====
 
 The equations we discussed above can be coded rather easily (the
-framework is similar to what we developed for logistic
-regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.
-!bc pycod
-
-!ec
-
-!split
-===== Problems with the Simpler Approach =====
+framework is similar to what we developed for logistic regression). We
+can set up a simple case with two classes only and we want to find a
+line which separates them the best possible way.
 
 
 There are however problems with this approach, although it looks
-pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
+pretty straightforward to implement. When running a code for such a
+case we can easily end up with many diffeent lines which separate the
+two classes.
 
 
 For small