diff --git a/doc/pub/svm/html/._svm-bs000.html b/doc/pub/svm/html/._svm-bs000.html index b7903aa11..43ca73433 100644 --- a/doc/pub/svm/html/._svm-bs000.html +++ b/doc/pub/svm/html/._svm-bs000.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • @@ -174,7 +174,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 23, 2019

    +

    Nov 25, 2019


    diff --git a/doc/pub/svm/html/._svm-bs001.html b/doc/pub/svm/html/._svm-bs001.html index 6382d89df..5775d787d 100644 --- a/doc/pub/svm/html/._svm-bs001.html +++ b/doc/pub/svm/html/._svm-bs001.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({

  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs002.html b/doc/pub/svm/html/._svm-bs002.html index 1751f46d8..3899f4b96 100644 --- a/doc/pub/svm/html/._svm-bs002.html +++ b/doc/pub/svm/html/._svm-bs002.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs003.html b/doc/pub/svm/html/._svm-bs003.html index f48b78c5a..5c6d9f26b 100644 --- a/doc/pub/svm/html/._svm-bs003.html +++ b/doc/pub/svm/html/._svm-bs003.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs004.html b/doc/pub/svm/html/._svm-bs004.html index 80a25a37c..79151bb47 100644 --- a/doc/pub/svm/html/._svm-bs004.html +++ b/doc/pub/svm/html/._svm-bs004.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs005.html b/doc/pub/svm/html/._svm-bs005.html index e173f9bc4..a8352e871 100644 --- a/doc/pub/svm/html/._svm-bs005.html +++ b/doc/pub/svm/html/._svm-bs005.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs006.html b/doc/pub/svm/html/._svm-bs006.html index f32cde5f5..d8b4917e9 100644 --- a/doc/pub/svm/html/._svm-bs006.html +++ b/doc/pub/svm/html/._svm-bs006.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs007.html b/doc/pub/svm/html/._svm-bs007.html index b66e1dc7f..76da7b704 100644 --- a/doc/pub/svm/html/._svm-bs007.html +++ b/doc/pub/svm/html/._svm-bs007.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs008.html b/doc/pub/svm/html/._svm-bs008.html index 4e45296b0..09a98c3ef 100644 --- a/doc/pub/svm/html/._svm-bs008.html +++ b/doc/pub/svm/html/._svm-bs008.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs009.html b/doc/pub/svm/html/._svm-bs009.html index 4bfee26d9..582f5f760 100644 --- a/doc/pub/svm/html/._svm-bs009.html +++ b/doc/pub/svm/html/._svm-bs009.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs010.html b/doc/pub/svm/html/._svm-bs010.html index 432f59530..67265b469 100644 --- a/doc/pub/svm/html/._svm-bs010.html +++ b/doc/pub/svm/html/._svm-bs010.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs011.html b/doc/pub/svm/html/._svm-bs011.html index 316db0caa..ef14ce568 100644 --- a/doc/pub/svm/html/._svm-bs011.html +++ b/doc/pub/svm/html/._svm-bs011.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs012.html b/doc/pub/svm/html/._svm-bs012.html index b600ced75..de1948915 100644 --- a/doc/pub/svm/html/._svm-bs012.html +++ b/doc/pub/svm/html/._svm-bs012.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • @@ -155,7 +155,7 @@ MathJax.Hub.Config({ -

    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 diff --git a/doc/pub/svm/html/._svm-bs013.html b/doc/pub/svm/html/._svm-bs013.html index 302ea6e65..361980a5b 100644 --- a/doc/pub/svm/html/._svm-bs013.html +++ b/doc/pub/svm/html/._svm-bs013.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({

  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • @@ -155,7 +155,7 @@ MathJax.Hub.Config({ -

    Adding the muliplier

    +

    Adding the Multiplier

    However, we can add to diff --git a/doc/pub/svm/html/._svm-bs014.html b/doc/pub/svm/html/._svm-bs014.html index 562fea828..921b3d936 100644 --- a/doc/pub/svm/html/._svm-bs014.html +++ b/doc/pub/svm/html/._svm-bs014.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({

  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • @@ -155,7 +155,7 @@ MathJax.Hub.Config({ -

    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], diff --git a/doc/pub/svm/html/._svm-bs015.html b/doc/pub/svm/html/._svm-bs015.html index fde86dc1a..2efddc322 100644 --- a/doc/pub/svm/html/._svm-bs015.html +++ b/doc/pub/svm/html/._svm-bs015.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs016.html b/doc/pub/svm/html/._svm-bs016.html index 496a4d385..cab82a909 100644 --- a/doc/pub/svm/html/._svm-bs016.html +++ b/doc/pub/svm/html/._svm-bs016.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs017.html b/doc/pub/svm/html/._svm-bs017.html index d3188a034..cabeab656 100644 --- a/doc/pub/svm/html/._svm-bs017.html +++ b/doc/pub/svm/html/._svm-bs017.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs018.html b/doc/pub/svm/html/._svm-bs018.html index 7f64bc0d4..5e43608d0 100644 --- a/doc/pub/svm/html/._svm-bs018.html +++ b/doc/pub/svm/html/._svm-bs018.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs019.html b/doc/pub/svm/html/._svm-bs019.html index 8c92dae33..7e80bff40 100644 --- a/doc/pub/svm/html/._svm-bs019.html +++ b/doc/pub/svm/html/._svm-bs019.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • @@ -158,11 +158,11 @@ MathJax.Hub.Config({

    Kernels and non-linearity

    -The cases we have studied till were all characterized by two classes +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 higher-order polynomials, +space using other basis expansions such as higher-order polynomials, wavelets, splines etc.

    diff --git a/doc/pub/svm/html/._svm-bs020.html b/doc/pub/svm/html/._svm-bs020.html index 372609da7..6840b7357 100644 --- a/doc/pub/svm/html/._svm-bs020.html +++ b/doc/pub/svm/html/._svm-bs020.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({

  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • @@ -175,7 +175,7 @@ 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 kerne \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ @@ -195,7 +195,7 @@ 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. +\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.

    diff --git a/doc/pub/svm/html/._svm-bs021.html b/doc/pub/svm/html/._svm-bs021.html index 814f24f67..e891756ce 100644 --- a/doc/pub/svm/html/._svm-bs021.html +++ b/doc/pub/svm/html/._svm-bs021.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({

  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs022.html b/doc/pub/svm/html/._svm-bs022.html index 425544721..52686017e 100644 --- a/doc/pub/svm/html/._svm-bs022.html +++ b/doc/pub/svm/html/._svm-bs022.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs023.html b/doc/pub/svm/html/._svm-bs023.html index 7523745b2..410c286f4 100644 --- a/doc/pub/svm/html/._svm-bs023.html +++ b/doc/pub/svm/html/._svm-bs023.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs024.html b/doc/pub/svm/html/._svm-bs024.html index 276edbbfc..311c05184 100644 --- a/doc/pub/svm/html/._svm-bs024.html +++ b/doc/pub/svm/html/._svm-bs024.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs025.html b/doc/pub/svm/html/._svm-bs025.html index e857d117b..3762e072b 100644 --- a/doc/pub/svm/html/._svm-bs025.html +++ b/doc/pub/svm/html/._svm-bs025.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/._svm-bs026.html b/doc/pub/svm/html/._svm-bs026.html index 09c865519..38647b2d0 100644 --- a/doc/pub/svm/html/._svm-bs026.html +++ b/doc/pub/svm/html/._svm-bs026.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • diff --git a/doc/pub/svm/html/svm-bs.html b/doc/pub/svm/html/svm-bs.html index b7903aa11..43ca73433 100644 --- a/doc/pub/svm/html/svm-bs.html +++ b/doc/pub/svm/html/svm-bs.html @@ -52,12 +52,12 @@ Automatically generated HTML file from DocOnce source ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -123,9 +123,9 @@ MathJax.Hub.Config({
  • Code Example
  • Problems with the Simpler Approach
  • A better approach
    • -
  • A quick reminder on Lagrangian multipliers
    • -
  • Adding the muliplier
    • -
  • Setting up the problem
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
  • The problem to solve
  • The last steps
  • A soft classifier
    • @@ -174,7 +174,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 23, 2019

    +

    Nov 25, 2019


    diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html index a4efba160..95fb022fa 100644 --- a/doc/pub/svm/html/svm-reveal.html +++ b/doc/pub/svm/html/svm-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Nov 23, 2019

    +

    Nov 25, 2019


    @@ -554,7 +554,7 @@ 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 @@ -616,7 +616,7 @@ Then \( dz \) is no longer arbitrary.

    -

    Adding the muliplier

    +

    Adding the Multiplier

    However, we can add to @@ -671,7 +671,7 @@ $$

    -

    Setting up the problem

    +

    Setting up the Problem

    In order to solve the above problem, we define the following Lagrangian function to be minimized

     
    $$ @@ -909,11 +909,11 @@ $$

    Kernels and non-linearity

    -The cases we have studied till were all characterized by two classes +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 higher-order polynomials, +space using other basis expansions such as higher-order polynomials, wavelets, splines etc.

    @@ -1006,7 +1006,7 @@ $$

     
    from which we also find \( b \). -To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kerne \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +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). @@ -1030,7 +1030,7 @@ 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. +\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.

    diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html index 722543128..3feffe35c 100644 --- a/doc/pub/svm/html/svm-solarized.html +++ b/doc/pub/svm/html/svm-solarized.html @@ -46,12 +46,12 @@ div { text-align: justify; text-justify: inter-word; } ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -110,7 +110,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 23, 2019

    +

    Nov 25, 2019












    @@ -472,7 +472,7 @@ 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 @@ -522,7 +522,7 @@ Then \( dz \) is no longer arbitrary.











    -

    Adding the muliplier

    +

    Adding the Multiplier

    However, we can add to @@ -565,7 +565,7 @@ $$











    -

    Setting up the problem

    +

    Setting up the Problem

    In order to solve the above problem, we define the following Lagrangian function to be minimized $$ {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], @@ -756,11 +756,11 @@ $$

    Kernels and non-linearity

    -The cases we have studied till were all characterized by two classes +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 higher-order polynomials, +space using other basis expansions such as higher-order polynomials, wavelets, splines etc.

    @@ -846,7 +846,7 @@ 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 kerne \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ @@ -866,7 +866,7 @@ 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. +\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.











    diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html index 8579530c4..d8cb22755 100644 --- a/doc/pub/svm/html/svm.html +++ b/doc/pub/svm/html/svm.html @@ -51,12 +51,12 @@ div { text-align: justify; text-justify: inter-word; } ('Code Example', 2, None, '___sec8'), ('Problems with the Simpler Approach', 2, None, '___sec9'), ('A better approach', 2, None, '___sec10'), - ('A quick reminder on Lagrangian multipliers', + ('A quick Reminder on Lagrangian Multipliers', 2, None, '___sec11'), - ('Adding the muliplier', 2, None, '___sec12'), - ('Setting up the problem', 2, None, '___sec13'), + ('Adding the Multiplier', 2, None, '___sec12'), + ('Setting up the Problem', 2, None, '___sec13'), ('The problem to solve', 2, None, '___sec14'), ('The last steps', 2, None, '___sec15'), ('A soft classifier', 2, None, '___sec16'), @@ -115,7 +115,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 23, 2019

    +

    Nov 25, 2019












    @@ -477,7 +477,7 @@ 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 @@ -527,7 +527,7 @@ Then \( dz \) is no longer arbitrary.











    -

    Adding the muliplier

    +

    Adding the Multiplier

    However, we can add to @@ -570,7 +570,7 @@ $$











    -

    Setting up the problem

    +

    Setting up the Problem

    In order to solve the above problem, we define the following Lagrangian function to be minimized $$ {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], @@ -761,11 +761,11 @@ $$

    Kernels and non-linearity

    -The cases we have studied till were all characterized by two classes +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 higher-order polynomials, +space using other basis expansions such as higher-order polynomials, wavelets, splines etc.

    @@ -851,7 +851,7 @@ 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 kerne \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ @@ -871,7 +871,7 @@ 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. +\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.











    diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz index b7c9245f6..d289e0cf1 100644 Binary files a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz and b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz differ diff --git a/doc/pub/svm/ipynb/svm.ipynb b/doc/pub/svm/ipynb/svm.ipynb index 181342b04..bfca7f0ea 100644 --- a/doc/pub/svm/ipynb/svm.ipynb +++ b/doc/pub/svm/ipynb/svm.ipynb @@ -10,7 +10,7 @@ " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Nov 23, 2019**\n", + "Date: **Nov 25, 2019**\n", "\n", "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -526,7 +526,7 @@ "possible margin $M$. Before we proceed, we need to remind ourselves\n", "about Lagrangian multipliers.\n", "\n", - "## A quick reminder on Lagrangian multipliers\n", + "## A quick Reminder on Lagrangian Multipliers\n", "\n", "Consider a function of three independent variables $f(x,y,z)$ . For the function $f$ to be an\n", "extreme we have" @@ -637,7 +637,7 @@ "variables.\n", "Then $dz$ is no longer arbitrary.\n", "\n", - "## Adding the muliplier\n", + "## Adding the Multiplier\n", "\n", "However, we can add to" ] @@ -741,7 +741,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Setting up the problem\n", + "## Setting up the Problem\n", "In order to solve the above problem, we define the following Lagrangian function to be minimized" ] }, @@ -1169,11 +1169,11 @@ "source": [ "## Kernels and non-linearity\n", "\n", - "The cases we have studied till were all characterized by two classes\n", + "The cases we have studied till now, 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", + "space using other basis expansions such as higher-order polynomials,\n", "wavelets, splines etc.\n", "\n", "If our feature space is not easy to separate, as shown in the figure\n", @@ -1296,7 +1296,7 @@ "metadata": {}, "source": [ "from which we also find $b$.\n", - "To compute $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we define the kerne $K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$ as" + "To compute $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we define the kernel $K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$ as" ] }, { @@ -1337,7 +1337,7 @@ "This leads to the so-called\n", "kernel trick and the result leads to the same as if we went through\n", "the trouble of performing the transformation\n", - "$(\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j)$ during the SVM calculations.\n", + "$\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j)$ during the SVM calculations.\n", "\n", "\n", "## The problem to solve\n", diff --git a/doc/pub/svm/pdf/svm-minted.pdf b/doc/pub/svm/pdf/svm-minted.pdf index 3f1b41988..c2b7b60d4 100644 Binary files a/doc/pub/svm/pdf/svm-minted.pdf and b/doc/pub/svm/pdf/svm-minted.pdf differ diff --git a/doc/src/SupportVMachines/svm.do.txt b/doc/src/SupportVMachines/svm.do.txt index ee7906ce9..e1977f9c2 100644 --- a/doc/src/SupportVMachines/svm.do.txt +++ b/doc/src/SupportVMachines/svm.do.txt @@ -342,7 +342,7 @@ possible margin $M$. Before we proceed, we need to remind ourselves about Lagrangian multipliers. !split -===== 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 extreme we have @@ -394,7 +394,7 @@ variables. Then $dz$ is no longer arbitrary. !split -===== Adding the muliplier ===== +===== Adding the Multiplier ===== However, we can add to !bt @@ -441,7 +441,7 @@ If we have a set of constraints $\phi_k$ we have the equations !et !split -===== Setting up the problem ===== +===== Setting up the Problem ===== In order to solve the above problem, we define the following Lagrangian function to be minimized !bt \[ @@ -634,11 +634,11 @@ y_i(\bm{w}^T\bm{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. !split ===== Kernels and non-linearity ===== -The cases we have studied till were all characterized by two classes +The cases we have studied till now, were all characterized by two classes with a close to linear separability. The classifiers we have described so far find linear boundaries in our input feature space. It is possible to make our procedure more flexible by exploring the feature -space using other basis expansions such higher-order polynomials, +space using other basis expansions such as higher-order polynomials, wavelets, splines etc. If our feature space is not easy to separate, as shown in the figure @@ -724,7 +724,7 @@ y_i(\bm{w}^T\bm{z}_i+b)= 1 \hspace{0.1cm}\forall i, \] !et from which we also find $b$. -To compute $\bm{z}_i^T\bm{z}_j$ we define the kerne $K(\bm{x}_i,\bm{x}_j)$ as +To compute $\bm{z}_i^T\bm{z}_j$ we define the kernel $K(\bm{x}_i,\bm{x}_j)$ as !bt \[ K(\bm{x}_i,\bm{x}_j)=\bm{z}_i^T\bm{z}_j= \phi(\bm{x}_i)^T\phi(\bm{x}_j). @@ -746,7 +746,7 @@ the dot product $(\bm{x}_i^T\bm{x}_j)^2$. 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 -$(\phi(\bm{x}_i)^T\phi(\bm{x}_j)$ during the SVM calculations. +$\phi(\bm{x}_i)^T\phi(\bm{x}_j)$ during the SVM calculations. !split