From 0fce698a6335282192ec4e9af42c5246c2080d77 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Mon, 5 Nov 2018 22:40:28 +0100 Subject: [PATCH] writing code for sim using cvxopt --- doc/pub/svm/html/._svm-bs000.html | 14 +- doc/pub/svm/html/._svm-bs001.html | 14 +- doc/pub/svm/html/._svm-bs002.html | 14 +- doc/pub/svm/html/._svm-bs003.html | 14 +- doc/pub/svm/html/._svm-bs004.html | 14 +- doc/pub/svm/html/._svm-bs005.html | 14 +- doc/pub/svm/html/._svm-bs006.html | 14 +- doc/pub/svm/html/._svm-bs007.html | 14 +- doc/pub/svm/html/._svm-bs008.html | 14 +- doc/pub/svm/html/._svm-bs009.html | 15 +- doc/pub/svm/html/._svm-bs010.html | 16 +- doc/pub/svm/html/._svm-bs011.html | 17 +- doc/pub/svm/html/._svm-bs012.html | 18 +- doc/pub/svm/html/._svm-bs013.html | 17 +- doc/pub/svm/html/._svm-bs014.html | 17 +- doc/pub/svm/html/._svm-bs015.html | 17 +- doc/pub/svm/html/._svm-bs016.html | 17 +- doc/pub/svm/html/._svm-bs017.html | 37 +++- doc/pub/svm/html/._svm-bs018.html | 207 ++++++++++++++++++++++ doc/pub/svm/html/._svm-bs019.html | 187 ++++++++++++++++++++ doc/pub/svm/html/._svm-bs020.html | 186 ++++++++++++++++++++ doc/pub/svm/html/._svm-bs021.html | 185 +++++++++++++++++++ doc/pub/svm/html/._svm-bs022.html | 234 +++++++++++++++++++++++++ doc/pub/svm/html/svm-bs.html | 14 +- doc/pub/svm/html/svm-reveal.html | 121 +++++++++++++ doc/pub/svm/html/svm-solarized.html | 121 ++++++++++++- doc/pub/svm/html/svm.html | 121 ++++++++++++- doc/pub/svm/ipynb/ipynb-svm-src.tar.gz | Bin 207 -> 206 bytes doc/pub/svm/ipynb/svm.ipynb | 153 +++++++++++++++- doc/pub/svm/pdf/svm-minted.pdf | Bin 243532 -> 298508 bytes doc/src/SupportVMachines/svm.do.txt | 95 ++++++++++ 31 files changed, 1890 insertions(+), 31 deletions(-) create mode 100644 doc/pub/svm/html/._svm-bs018.html create mode 100644 doc/pub/svm/html/._svm-bs019.html create mode 100644 doc/pub/svm/html/._svm-bs020.html create mode 100644 doc/pub/svm/html/._svm-bs021.html create mode 100644 doc/pub/svm/html/._svm-bs022.html diff --git a/doc/pub/svm/html/._svm-bs000.html b/doc/pub/svm/html/._svm-bs000.html index ece869c15..9fe71862a 100644 --- a/doc/pub/svm/html/._svm-bs000.html +++ b/doc/pub/svm/html/._svm-bs000.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -172,7 +182,7 @@ MathJax.Hub.Config({
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs001.html b/doc/pub/svm/html/._svm-bs001.html index 5c9c4aa14..6c0d0f6bf 100644 --- a/doc/pub/svm/html/._svm-bs001.html +++ b/doc/pub/svm/html/._svm-bs001.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -171,7 +181,7 @@ We distinguish also between linear and non-linear approaches. The latter are the
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    • diff --git a/doc/pub/svm/html/._svm-bs002.html b/doc/pub/svm/html/._svm-bs002.html index 06496c435..c17a6cd20 100644 --- a/doc/pub/svm/html/._svm-bs002.html +++ b/doc/pub/svm/html/._svm-bs002.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -162,7 +172,7 @@ circles.
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    • diff --git a/doc/pub/svm/html/._svm-bs003.html b/doc/pub/svm/html/._svm-bs003.html index 4ce42b1ac..f159bd84e 100644 --- a/doc/pub/svm/html/._svm-bs003.html +++ b/doc/pub/svm/html/._svm-bs003.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -172,7 +182,7 @@ $$
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    • diff --git a/doc/pub/svm/html/._svm-bs004.html b/doc/pub/svm/html/._svm-bs004.html index 37db97a20..79f6ee766 100644 --- a/doc/pub/svm/html/._svm-bs004.html +++ b/doc/pub/svm/html/._svm-bs004.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -188,7 +198,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
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  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -174,7 +184,7 @@ for our data sample.
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  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -170,7 +180,7 @@ $$
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    • diff --git a/doc/pub/svm/html/._svm-bs007.html b/doc/pub/svm/html/._svm-bs007.html index 148601449..685685aef 100644 --- a/doc/pub/svm/html/._svm-bs007.html +++ b/doc/pub/svm/html/._svm-bs007.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -174,7 +184,7 @@ $$
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    • diff --git a/doc/pub/svm/html/._svm-bs008.html b/doc/pub/svm/html/._svm-bs008.html index b650a6265..cee105910 100644 --- a/doc/pub/svm/html/._svm-bs008.html +++ b/doc/pub/svm/html/._svm-bs008.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -177,6 +187,8 @@ at all.
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    • diff --git a/doc/pub/svm/html/._svm-bs009.html b/doc/pub/svm/html/._svm-bs009.html index e0042ab0d..00e3f8aa0 100644 --- a/doc/pub/svm/html/._svm-bs009.html +++ b/doc/pub/svm/html/._svm-bs009.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -188,6 +198,9 @@ We have thus defined our margin as the invers of the norm of \( \boldsymbol{w} \
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  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -199,6 +209,10 @@ Then \( dz \) is no longer arbitrary.
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    • diff --git a/doc/pub/svm/html/._svm-bs011.html b/doc/pub/svm/html/._svm-bs011.html index 4bf7ad8b7..9941de0ab 100644 --- a/doc/pub/svm/html/._svm-bs011.html +++ b/doc/pub/svm/html/._svm-bs011.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -191,6 +201,11 @@ $$
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    • diff --git a/doc/pub/svm/html/._svm-bs012.html b/doc/pub/svm/html/._svm-bs012.html index 5274b124e..98c4ecd16 100644 --- a/doc/pub/svm/html/._svm-bs012.html +++ b/doc/pub/svm/html/._svm-bs012.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -188,6 +198,12 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
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    • diff --git a/doc/pub/svm/html/._svm-bs013.html b/doc/pub/svm/html/._svm-bs013.html index 230eacbcd..8ad3d9fc5 100644 --- a/doc/pub/svm/html/._svm-bs013.html +++ b/doc/pub/svm/html/._svm-bs013.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -170,6 +180,11 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
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    • diff --git a/doc/pub/svm/html/._svm-bs014.html b/doc/pub/svm/html/._svm-bs014.html index 690035116..f4042b91c 100644 --- a/doc/pub/svm/html/._svm-bs014.html +++ b/doc/pub/svm/html/._svm-bs014.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -179,6 +189,11 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs015.html b/doc/pub/svm/html/._svm-bs015.html index 6ee62d029..afa001e3c 100644 --- a/doc/pub/svm/html/._svm-bs015.html +++ b/doc/pub/svm/html/._svm-bs015.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -179,6 +189,11 @@ misclassifications.
  • 16
  • 17
  • 18
    • +
  • 19
    • +
  • 20
    • +
  • 21
    • +
  • 22
    • +
  • 23
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs016.html b/doc/pub/svm/html/._svm-bs016.html index eafe9c5b7..8a915f0c2 100644 --- a/doc/pub/svm/html/._svm-bs016.html +++ b/doc/pub/svm/html/._svm-bs016.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -197,6 +207,11 @@ $$
  • 16
  • 17
  • 18
    • +
  • 19
    • +
  • 20
    • +
  • 21
    • +
  • 22
    • +
  • 23
  • »
    • diff --git a/doc/pub/svm/html/._svm-bs017.html b/doc/pub/svm/html/._svm-bs017.html index 68036b41f..5486edbe8 100644 --- a/doc/pub/svm/html/._svm-bs017.html +++ b/doc/pub/svm/html/._svm-bs017.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -131,6 +141,25 @@ MathJax.Hub.Config({

    Kernels and non-linearity

    +

    +The cases we have studied till 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 higher-order polynomials, +wavelets, splines etc. + +

    +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. + +

    +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. + +

    diff --git a/doc/pub/svm/html/._svm-bs018.html b/doc/pub/svm/html/._svm-bs018.html new file mode 100644 index 000000000..0a17b8423 --- /dev/null +++ b/doc/pub/svm/html/._svm-bs018.html @@ -0,0 +1,207 @@ + + + + + + + +Data Analysis and Machine Learning: Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Support Vector Machines +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    The equations

    + +

    +Suppose we define a polynomial transformation of degree two (we continue to live in a plane with \( x_1 \) and \( x_2 \) as variables) +$$ +z = \phi(x) =\left(1, x_1, x_2, x_1^2, x_2^2, x_1x_2). +$$ + +

    +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 \( b \). + +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/svm/html/._svm-bs019.html b/doc/pub/svm/html/._svm-bs019.html new file mode 100644 index 000000000..af36a2942 --- /dev/null +++ b/doc/pub/svm/html/._svm-bs019.html @@ -0,0 +1,187 @@ + + + + + + + +Data Analysis and Machine Learning: Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Support Vector Machines +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Different kernels

    + +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/svm/html/._svm-bs020.html b/doc/pub/svm/html/._svm-bs020.html new file mode 100644 index 000000000..ddf6ab472 --- /dev/null +++ b/doc/pub/svm/html/._svm-bs020.html @@ -0,0 +1,186 @@ + + + + + + + +Data Analysis and Machine Learning: Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Support Vector Machines +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Quadratic coefficient matrix

    + +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/svm/html/._svm-bs021.html b/doc/pub/svm/html/._svm-bs021.html new file mode 100644 index 000000000..d5d695472 --- /dev/null +++ b/doc/pub/svm/html/._svm-bs021.html @@ -0,0 +1,185 @@ + + + + + + + +Data Analysis and Machine Learning: Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Support Vector Machines +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Mercer's theorem

    + +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/svm/html/._svm-bs022.html b/doc/pub/svm/html/._svm-bs022.html new file mode 100644 index 000000000..0325d756e --- /dev/null +++ b/doc/pub/svm/html/._svm-bs022.html @@ -0,0 +1,234 @@ + + + + + + + +Data Analysis and Machine Learning: Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Support Vector Machines +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    How do we solve these problems

    + +

    +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. + +

    +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it +

    + + +

    import numpy
+import cvxopt
+
    +

    +Let us first set up the standard form the of quadratic programming (QP) equations by defining the problem as +$$ +\mathrm{min} +$$ + +

    +subject to Gx u. Note that x itself is not provided to the solver, since it is an internal +variable being optimized over. In particular, this means that the solver has no explicit knowledge +of x itself; everything is implicity defined by the supplied parameters. It is essential +that the same variable order is maintained for the relevant parameters (e.g., qi +Non-convexity implies the existence of local optima, making it difficult to find global optima. + +

    +collapsed all inequality constraints into a single G matrix of the standard form. +Since there are no equality constraints, we do not need to provide the empty A, b. Note +that even though y +2 did not appear in the original objective, we had to include it with zero +coefficients in P because the solver parameters must be defined using the full set of variables. +Even if certain variables only appear in constraints, they will still need to be expressed with +zero coefficients in the objective parameters, and vice versa. +Let us first define the above parameters in Python. CVXOPT supplies its own matrix +object; all arguments given to its solvers must be in this matrix type. There are two ways +to do this. The first is to define the matrix directly with (potentially nested) lists: +from cvxopt import matrix +

    + + +

    P = matrix([[1.0,0.0],[0.0,0.0]])
+q = matrix([3.0,4.0])
+G = matrix([[-1.0,0.0,-1.0,2.0,3.0],[0.0,-1.0,-3.0,5.0,4.0]])
+h = matrix([0.0,0.0,-15.0,100.0,80.0])
+
    +

    + +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/svm/html/svm-bs.html b/doc/pub/svm/html/svm-bs.html index ece869c15..9fe71862a 100644 --- a/doc/pub/svm/html/svm-bs.html +++ b/doc/pub/svm/html/svm-bs.html @@ -59,7 +59,12 @@ Automatically generated HTML file from DocOnce source ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -114,6 +119,11 @@ MathJax.Hub.Config({
  • A soft classifier
  • Soft optmization problem
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • Different kernels
    • +
  • Quadratic coefficient matrix
    • +
  • Mercer's theorem
    • +
  • How do we solve these problems
    • @@ -172,7 +182,7 @@ MathJax.Hub.Config({
  • 9
  • 10
  • ...
    • -
  • 18
    • +
  • 23
  • »
    • diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html index 5243d34be..dc4886414 100644 --- a/doc/pub/svm/html/svm-reveal.html +++ b/doc/pub/svm/html/svm-reveal.html @@ -804,6 +804,127 @@ $$

    Kernels and non-linearity

    + +

    +The cases we have studied till 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 higher-order polynomials, +wavelets, splines etc. + +

    +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. + +

    +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. +

    + + +
    +

    The equations

    + +

    +Suppose we define a polynomial transformation of degree two (we continue to live in a plane with \( x_1 \) and \( x_2 \) as variables) +

     
    +$$ +z = \phi(x) =\left(1, x_1, x_2, x_1^2, x_2^2, x_1x_2). +$$ +

     
    + +

    +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 \( b \). +

    + + +
    +

    Different kernels

    +
    + + +
    +

    Quadratic coefficient matrix

    +
    + + +
    +

    Mercer's theorem

    +
    + + +
    +

    How do we solve these problems

    + +

    +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. + +

    +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it +

    + + +

    import numpy
+import cvxopt
+
    +

    +Let us first set up the standard form the of quadratic programming (QP) equations by defining the problem as +

     
    +$$ +\mathrm{min} +$$ +

     
    + +

    +subject to Gx u. Note that x itself is not provided to the solver, since it is an internal +variable being optimized over. In particular, this means that the solver has no explicit knowledge +of x itself; everything is implicity defined by the supplied parameters. It is essential +that the same variable order is maintained for the relevant parameters (e.g., qi +Non-convexity implies the existence of local optima, making it difficult to find global optima. + +

    +collapsed all inequality constraints into a single G matrix of the standard form. +Since there are no equality constraints, we do not need to provide the empty A, b. Note +that even though y +2 did not appear in the original objective, we had to include it with zero +coefficients in P because the solver parameters must be defined using the full set of variables. +Even if certain variables only appear in constraints, they will still need to be expressed with +zero coefficients in the objective parameters, and vice versa. +Let us first define the above parameters in Python. CVXOPT supplies its own matrix +object; all arguments given to its solvers must be in this matrix type. There are two ways +to do this. The first is to define the matrix directly with (potentially nested) lists: +from cvxopt import matrix +

    + + +

    P = matrix([[1.0,0.0],[0.0,0.0]])
+q = matrix([3.0,4.0])
+G = matrix([[-1.0,0.0,-1.0,2.0,3.0],[0.0,-1.0,-3.0,5.0,4.0]])
+h = matrix([0.0,0.0,-15.0,100.0,80.0])
+
    diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html index 85221cb67..9f39b8eff 100644 --- a/doc/pub/svm/html/svm-solarized.html +++ b/doc/pub/svm/html/svm-solarized.html @@ -53,7 +53,12 @@ div { text-align: justify; text-justify: inter-word; } ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -639,6 +644,120 @@ $$

    Kernels and non-linearity

    +

    +The cases we have studied till 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 higher-order polynomials, +wavelets, splines etc. + +

    +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. + +

    +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. + +

    +









    + +

    The equations

    + +

    +Suppose we define a polynomial transformation of degree two (we continue to live in a plane with \( x_1 \) and \( x_2 \) as variables) +$$ +z = \phi(x) =\left(1, x_1, x_2, x_1^2, x_2^2, x_1x_2). +$$ + +

    +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 \( b \). + +

    +









    + +

    Different kernels

    + +

    +









    + +

    Quadratic coefficient matrix

    + +

    + + +

    Mercer's theorem

    + +

    +









    + +

    How do we solve these problems

    + +

    +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. + +

    +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it +

    + + +

    import numpy
+import cvxopt
+
    +

    +Let us first set up the standard form the of quadratic programming (QP) equations by defining the problem as +$$ +\mathrm{min} +$$ + +

    +subject to Gx u. Note that x itself is not provided to the solver, since it is an internal +variable being optimized over. In particular, this means that the solver has no explicit knowledge +of x itself; everything is implicity defined by the supplied parameters. It is essential +that the same variable order is maintained for the relevant parameters (e.g., qi +Non-convexity implies the existence of local optima, making it difficult to find global optima. + +

    +collapsed all inequality constraints into a single G matrix of the standard form. +Since there are no equality constraints, we do not need to provide the empty A, b. Note +that even though y +2 did not appear in the original objective, we had to include it with zero +coefficients in P because the solver parameters must be defined using the full set of variables. +Even if certain variables only appear in constraints, they will still need to be expressed with +zero coefficients in the objective parameters, and vice versa. +Let us first define the above parameters in Python. CVXOPT supplies its own matrix +object; all arguments given to its solvers must be in this matrix type. There are two ways +to do this. The first is to define the matrix directly with (potentially nested) lists: +from cvxopt import matrix +

    + + +

    P = matrix([[1.0,0.0],[0.0,0.0]])
+q = matrix([3.0,4.0])
+G = matrix([[-1.0,0.0,-1.0,2.0,3.0],[0.0,-1.0,-3.0,5.0,4.0]])
+h = matrix([0.0,0.0,-15.0,100.0,80.0])
+
    +

    + diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html index 4d0d9e0e2..c9aca2c61 100644 --- a/doc/pub/svm/html/svm.html +++ b/doc/pub/svm/html/svm.html @@ -58,7 +58,12 @@ div { text-align: justify; text-justify: inter-word; } ('The last steps', 2, None, '___sec13'), ('A soft classifier', 2, None, '___sec14'), ('Soft optmization problem', 2, None, '___sec15'), - ('Kernels and non-linearity', 2, None, '___sec16')]} + ('Kernels and non-linearity', 2, None, '___sec16'), + ('The equations', 2, None, '___sec17'), + ('Different kernels', 2, None, '___sec18'), + ('Quadratic coefficient matrix', 2, None, '___sec19'), + ("Mercer's theorem", 2, None, '___sec20'), + ('How do we solve these problems', 2, None, '___sec21')]} end of tocinfo --> @@ -644,6 +649,120 @@ $$

    Kernels and non-linearity

    +

    +The cases we have studied till 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 higher-order polynomials, +wavelets, splines etc. + +

    +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. + +

    +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. + +

    +









    + +

    The equations

    + +

    +Suppose we define a polynomial transformation of degree two (we continue to live in a plane with \( x_1 \) and \( x_2 \) as variables) +$$ +z = \phi(x) =\left(1, x_1, x_2, x_1^2, x_2^2, x_1x_2). +$$ + +

    +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 \( b \). + +

    +









    + +

    Different kernels

    + +

    +









    + +

    Quadratic coefficient matrix

    + +

    + + +

    Mercer's theorem

    + +

    +









    + +

    How do we solve these problems

    + +

    +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. + +

    +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it +

    + + +

    import numpy
+import cvxopt
+
    +

    +Let us first set up the standard form the of quadratic programming (QP) equations by defining the problem as +$$ +\mathrm{min} +$$ + +

    +subject to Gx u. Note that x itself is not provided to the solver, since it is an internal +variable being optimized over. In particular, this means that the solver has no explicit knowledge +of x itself; everything is implicity defined by the supplied parameters. It is essential +that the same variable order is maintained for the relevant parameters (e.g., qi +Non-convexity implies the existence of local optima, making it difficult to find global optima. + +

    +collapsed all inequality constraints into a single G matrix of the standard form. +Since there are no equality constraints, we do not need to provide the empty A, b. Note +that even though y +2 did not appear in the original objective, we had to include it with zero +coefficients in P because the solver parameters must be defined using the full set of variables. +Even if certain variables only appear in constraints, they will still need to be expressed with +zero coefficients in the objective parameters, and vice versa. +Let us first define the above parameters in Python. CVXOPT supplies its own matrix +object; all arguments given to its solvers must be in this matrix type. There are two ways +to do this. The first is to define the matrix directly with (potentially nested) lists: +from cvxopt import matrix +

    + + +

    P = matrix([[1.0,0.0],[0.0,0.0]])
+q = matrix([3.0,4.0])
+G = matrix([[-1.0,0.0,-1.0,2.0,3.0],[0.0,-1.0,-3.0,5.0,4.0]])
+h = matrix([0.0,0.0,-15.0,100.0,80.0])
+
    +

    + diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz index c0c08712c1cde829667ea589713cbb6031f14e8d..a514400d1bac340f342b9a9c75444fe8be62633a 100644 GIT binary patch literal 206 zcmb2|=3v;c<3Th7^V{>jT}*}oZHd?CE%Ljtq<&c}e-F2KQ-^p*$rj_RMV?b$-u7Q@ zu&zX1>9gJJ^+H{Xvh4o-nbo)X%VakH#Zy+T_Bv;|%3g0}*X!R(=ebXwE?V+wlU9~$ z?YE;2S3ZaBvOKoLdB4~1+!&L0dGYtRMcK|MwsrmY$yX;zI@RcP(CfJt^UqxR-FxJ+ z-sa!ubeDcqoA+OJ#^H->%C|}v%|52~dhR{G?+4d6-#eP~2T~mMke~$FAZMW;c#l%mXUnB7PoNA=b?>nseb}73#AI4tF zjNjq>-ux2V>~po9-#))sB>K48F!%K)?t3*o{Dym6H~Sy`ac@2YGB{BDlda@tw;O{7 G0|NjwtzxbK diff --git a/doc/pub/svm/ipynb/svm.ipynb b/doc/pub/svm/ipynb/svm.ipynb index 346e18da1..cd34fab3c 100644 --- a/doc/pub/svm/ipynb/svm.ipynb +++ b/doc/pub/svm/ipynb/svm.ipynb @@ -1061,7 +1061,158 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Kernels and non-linearity" + "## Kernels and non-linearity\n", + "\n", + "The cases we have studied till were all characterized by two classes\n", + "with a close to linear separability. The classifiers we have described\n", + "so far find linear boundaries in our input feature space. It is\n", + "possible to make our procedure more flexible by exploring the feature\n", + "space using other basis expansions such higher-order polynomials,\n", + "wavelets, splines etc.\n", + "\n", + "If our feature space is not easy to separate, as shown in the figure\n", + "here, we can achieve a better separation by introducing more complex\n", + "basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to \n", + "obtain a separation between the classes which is almost linear. \n", + "\n", + "The change of basis, from $x\\rightarrow z=\\phi(x)$ leads to the same type of equations to be solved, except that\n", + "we need to introduce for example a polynomial transformation to a two-dimensional training set.\n", + "\n", + "## The equations\n", + "\n", + "Suppose we define a polynomial transformation of degree two (we continue to live in a plane with $x_1$ and $x_2$ as variables)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "z = \\phi(x) =\\left(1, x_1, x_2, x_1^2, x_2^2, x_1x_2).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{z}_i^T\\boldsymbol{Z}_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$, and for the support vectors" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y_i(\\boldsymbol{w}^T\\boldsymbol{z}_i+b)= 1 \\hspace{0.1cm}\\forall i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "from which we also find $b$. \n", + "\n", + "## Different kernels\n", + "\n", + "## Quadratic coefficient matrix\n", + "\n", + "\n", + "## Mercer's theorem\n", + "\n", + "## How do we solve these problems\n", + "\n", + "If we use Python as programming language and wish to venture beyond\n", + "**scikit-learn**, **tensorflow** and similar software which makes our\n", + "lives so much easier, we need to dive into the wonderful world of\n", + "quadratic programming. We can, if we wish, solve the minimization\n", + "problem using say standard gradient methods or conjugate gradient\n", + "methods. However, these methods tend to exhibit a rather slow\n", + "converge. So, welcome to the promised land of quadratic programming.\n", + "\n", + "The functions we need are contained in the quadratic programming package **CVXOPT** and we need to import it" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy\n", + "import cvxopt" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us first set up the standard form the of quadratic programming (QP) equations by defining the problem as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{min}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "subject to Gx u. Note that x itself is not provided to the solver, since it is an internal\n", + "variable being optimized over. In particular, this means that the solver has no explicit knowledge\n", + "of x itself; everything is implicity defined by the supplied parameters. It is essential\n", + "that the same variable order is maintained for the relevant parameters (e.g., qi\n", + "Non-convexity implies the existence of local optima, making it difficult to find global optima.\n", + "\n", + "collapsed all inequality constraints into a single G matrix of the standard form.\n", + "Since there are no equality constraints, we do not need to provide the empty A, b. Note\n", + "that even though y\n", + "2 did not appear in the original objective, we had to include it with zero\n", + "coefficients in P because the solver parameters must be defined using the full set of variables.\n", + "Even if certain variables only appear in constraints, they will still need to be expressed with\n", + "zero coefficients in the objective parameters, and vice versa.\n", + "Let us first define the above parameters in Python. CVXOPT supplies its own matrix\n", + "object; all arguments given to its solvers must be in this matrix type. There are two ways\n", + "to do this. The first is to define the matrix directly with (potentially nested) lists:\n", + "from cvxopt import matrix" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "P = matrix([[1.0,0.0],[0.0,0.0]])\n", + "q = matrix([3.0,4.0])\n", + "G = matrix([[-1.0,0.0,-1.0,2.0,3.0],[0.0,-1.0,-3.0,5.0,4.0]])\n", + "h = matrix([0.0,0.0,-15.0,100.0,80.0])" ] } ], diff --git a/doc/pub/svm/pdf/svm-minted.pdf b/doc/pub/svm/pdf/svm-minted.pdf index 2785509193b406d66694d07a2d40bf1c01ea152a..2276b146f69c62ed6576fe48993c9af6b5ce1ceb 100644 GIT binary patch delta 183885 zcmV)DK*7Jv?hcHc5|AYUIFmp#D3MMNfBDkOK5aH}9G7>KiYN9Vn*&G9o@Sz!I-<0* zlV2Zzfci1hQ?5$oB9b6Tg20CdC9}s&W`BGqf1~}~{kw0z*D6!B(5kfA{Vr3s5_My; zsw{=oO?JP{{+|Ed&2Ceey3F4Xu78dp(#Q>4>U?zF#;ANB22X^ayqktFJbt_Rf5-ig zSt&)O>#WeVFfH%-^Kv`_ugb@1rZvB8T9f;39;bEFjml5^M%?Mu4M2RIwW6(bh0=rY zRnumLZG}-K9rO_n@B?hwnEZ!-!eC$g=!Xa-*bvrNStZ)qH0kzA3u|>|M5QWPu9&U8 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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 higher-order polynomials, +wavelets, splines etc. + +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. + +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. + +!split +===== The equations ===== + +Suppose we define a polynomial transformation of degree two (we continue to live in a plane with $x_1$ and $x_2$ as variables) +!bt +\[ +z = \phi(x) =\left(1, x_1, x_2, x_1^2, x_2^2, x_1x_2). +\] +!et + +With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) +!bt +\[ +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\bm{z}_i^T\bm{Z}_j, +\] +!et +subject to the constraints $\lambda_i\geq 0$, $\sum_i\lambda_iy_i=0$, and for the support vectors +!bt +\[ +y_i(\bm{w}^T\bm{z}_i+b)= 1 \hspace{0.1cm}\forall i, +\] +!et +from which we also find $b$. + +!split +===== Different kernels ===== + +!split +===== Quadratic coefficient matrix ===== + +!split +===== Mercer's theorem ===== + +!split +===== How do we solve these problems ===== + +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. + +The functions we need are contained in the quadratic programming package _CVXOPT_ and we need to import it +!bc pycod +import numpy +import cvxopt +!ec + +Let us first set up the standard form the of quadratic programming (QP) equations by defining the problem as +!bt +\[ +\mathrm{min} +\] +!et + +subject to Gx u. Note that x itself is not provided to the solver, since it is an internal +variable being optimized over. In particular, this means that the solver has no explicit knowledge +of x itself; everything is implicity defined by the supplied parameters. It is essential +that the same variable order is maintained for the relevant parameters (e.g., qi +Non-convexity implies the existence of local optima, making it difficult to find global optima. + +collapsed all inequality constraints into a single G matrix of the standard form. +Since there are no equality constraints, we do not need to provide the empty A, b. Note +that even though y +2 did not appear in the original objective, we had to include it with zero +coefficients in P because the solver parameters must be defined using the full set of variables. +Even if certain variables only appear in constraints, they will still need to be expressed with +zero coefficients in the objective parameters, and vice versa. +Let us first define the above parameters in Python. CVXOPT supplies its own matrix +object; all arguments given to its solvers must be in this matrix type. There are two ways +to do this. The first is to define the matrix directly with (potentially nested) lists: +from cvxopt import matrix +!bc pycod +P = matrix([[1.0,0.0],[0.0,0.0]]) +q = matrix([3.0,4.0]) +G = matrix([[-1.0,0.0,-1.0,2.0,3.0],[0.0,-1.0,-3.0,5.0,4.0]]) +h = matrix([0.0,0.0,-15.0,100.0,80.0]) +!ec