From 137a1edb6fd674903d5af57a844d5920917f43c1 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Thu, 25 Nov 2021 06:51:30 +0100 Subject: [PATCH] corrected slides --- doc/pub/week47/html/week47-bs.html | 140 +++--- doc/pub/week47/html/week47-reveal.html | 37 +- doc/pub/week47/html/week47-solarized.html | 37 +- doc/pub/week47/html/week47.html | 37 +- doc/pub/week47/ipynb/ipynb-week47-src.tar.gz | Bin 823750 -> 823750 bytes doc/pub/week47/ipynb/week47.ipynb | 488 ++++++++++--------- doc/src/week47/week47.do.txt | 18 +- 7 files changed, 345 insertions(+), 412 deletions(-) diff --git a/doc/pub/week47/html/week47-bs.html b/doc/pub/week47/html/week47-bs.html index ea9e55a0f..527cec9d1 100644 --- a/doc/pub/week47/html/week47-bs.html +++ b/doc/pub/week47/html/week47-bs.html @@ -54,7 +54,6 @@ doconce format html week47.do.txt --html_style=bootstrap --pygments_html_style=d None, 'first-attempt-at-a-minimization-approach'), ('Solving the equations', 2, None, 'solving-the-equations'), - ('Code Example', 2, None, 'code-example'), ('Problems with the Simpler Approach', 2, None, @@ -273,75 +272,74 @@ MathJax.Hub.Config({
  • Getting into the details
  • First attempt at a minimization approach
  • Solving the equations
    • -
  • Code Example
    • -
  • Problems with the Simpler Approach
    • -
  • A better approach
    • -
  • A quick Reminder on Lagrangian Multipliers
    • -
  • Adding the Multiplier
    • -
  • Setting up the Problem
    • -
  • The problem to solve
    • -
  • The last steps
    • -
  • A soft classifier
    • -
  • Soft optmization problem
    • -
  • Kernels and non-linearity
    • -
  • The equations
    • -
  • The problem to solve
    • -
  • Different kernels and Mercer's theorem
    • -
  • The moons example
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simple example
    • -
  • Back to the more realistic cases
    • -
  • Summary of course
    • -
  • What? Me worry? No final exam in this course!
    • -
  • What is the link between Artificial Intelligence and Machine Learning and some general Remarks
    • -
  • Going back to the beginning of the semester
    • -
  • Not so sharp distinctions
    • -
  • Topics we have covered this year
    • -
  • Statistical analysis and optimization of data
    • -
  • Machine learning
    • -
  • Learning outcomes and overarching aims of this course
    • -
  • Perspective on Machine Learning
    • -
  • Machine Learning Research
    • -
  • Starting your Machine Learning Project
    • -
  • Choose a Model and Algorithm
    • -
  • Preparing Your Data
    • -
  • Which Activation and Weights to Choose in Neural Networks
    • -
  • Optimization Methods and Hyperparameters
    • -
  • Resampling
    • -
  • Other courses on Data science and Machine Learning at UiO
    • -
  • Additional courses of interest
    • -
  • What's the future like?
    • -
  • Types of Machine Learning, a repetition
    • -
  • Why Boltzmann machines?
    • -
  • Boltzmann Machines
    • -
  • Some similarities and differences from DNNs
    • -
  • Boltzmann machines (BM)
    • -
  • A standard BM setup
    • -
  • The structure of the RBM network
    • -
  • The network
    • -
  • Goals
    • -
  • Joint distribution
    • -
  • Network Elements, the energy function
    • -
  • Defining different types of RBMs
    • -
  • More about RBMs
    • -
  • Autoencoders: Overarching view
    • -
  • Bayesian Machine Learning
    • -
  • Reinforcement Learning
    • -
  • Transfer learning
    • -
  • Adversarial learning
    • -
  • Dual learning
    • -
  • Distributed machine learning
    • -
  • Meta learning
    • -
  • The Challenges Facing Machine Learning
    • -
  • Explainable machine learning
    • -
  • Quantum machine learning
    • -
  • Quantum machine learning algorithms based on linear algebra
    • -
  • Quantum reinforcement learning
    • -
  • Quantum deep learning
    • -
  • Social machine learning
    • -
  • The last words?
    • -
  • Best wishes to you all and thanks so much for your heroic efforts this semester
    • +
  • Problems with the Simpler Approach
    • +
  • A better approach
    • +
  • A quick Reminder on Lagrangian Multipliers
    • +
  • Adding the Multiplier
    • +
  • Setting up the Problem
    • +
  • The problem to solve
    • +
  • The last steps
    • +
  • A soft classifier
    • +
  • Soft optmization problem
    • +
  • Kernels and non-linearity
    • +
  • The equations
    • +
  • The problem to solve
    • +
  • Different kernels and Mercer's theorem
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simple example
    • +
  • Back to the more realistic cases
    • +
  • Summary of course
    • +
  • What? Me worry? No final exam in this course!
    • +
  • What is the link between Artificial Intelligence and Machine Learning and some general Remarks
    • +
  • Going back to the beginning of the semester
    • +
  • Not so sharp distinctions
    • +
  • Topics we have covered this year
    • +
  • Statistical analysis and optimization of data
    • +
  • Machine learning
    • +
  • Learning outcomes and overarching aims of this course
    • +
  • Perspective on Machine Learning
    • +
  • Machine Learning Research
    • +
  • Starting your Machine Learning Project
    • +
  • Choose a Model and Algorithm
    • +
  • Preparing Your Data
    • +
  • Which Activation and Weights to Choose in Neural Networks
    • +
  • Optimization Methods and Hyperparameters
    • +
  • Resampling
    • +
  • Other courses on Data science and Machine Learning at UiO
    • +
  • Additional courses of interest
    • +
  • What's the future like?
    • +
  • Types of Machine Learning, a repetition
    • +
  • Why Boltzmann machines?
    • +
  • Boltzmann Machines
    • +
  • Some similarities and differences from DNNs
    • +
  • Boltzmann machines (BM)
    • +
  • A standard BM setup
    • +
  • The structure of the RBM network
    • +
  • The network
    • +
  • Goals
    • +
  • Joint distribution
    • +
  • Network Elements, the energy function
    • +
  • Defining different types of RBMs
    • +
  • More about RBMs
    • +
  • Autoencoders: Overarching view
    • +
  • Bayesian Machine Learning
    • +
  • Reinforcement Learning
    • +
  • Transfer learning
    • +
  • Adversarial learning
    • +
  • Dual learning
    • +
  • Distributed machine learning
    • +
  • Meta learning
    • +
  • The Challenges Facing Machine Learning
    • +
  • Explainable machine learning
    • +
  • Quantum machine learning
    • +
  • Quantum machine learning algorithms based on linear algebra
    • +
  • Quantum reinforcement learning
    • +
  • Quantum deep learning
    • +
  • Social machine learning
    • +
  • The last words?
    • +
  • Best wishes to you all and thanks so much for your heroic efforts this semester
    • @@ -396,7 +394,7 @@ MathJax.Hub.Config({
  • 9
  • 10
  • ...
    • -
  • 79
    • +
  • 78
  • »
    • diff --git a/doc/pub/week47/html/week47-reveal.html b/doc/pub/week47/html/week47-reveal.html index 008b67824..d0dcfb6ea 100644 --- a/doc/pub/week47/html/week47-reveal.html +++ b/doc/pub/week47/html/week47-reveal.html @@ -379,6 +379,8 @@ $$ \boldsymbol{x}^T\boldsymbol{w}+b=0. $$

     
    + +

    For figures, see handwritten notes for Thursday November 25.

    @@ -469,7 +471,9 @@ f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0, $$

     
    -

    as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.

    +

    as the function that determines the line \( L \) that separates two classes (our two features), see the figures in the handwritten notes for Thursday November 25. +. +

    Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).

    @@ -531,39 +535,13 @@ $$
    -

    Code Example

    +

    Problems with the Simpler Approach

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

    - -
    -
    -
    -
    -
    - 
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -
    -

    Problems with the Simpler Approach

    -

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

    @@ -743,6 +721,7 @@ $$

    Setting up the Problem

    +

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

     
    $$ diff --git a/doc/pub/week47/html/week47-solarized.html b/doc/pub/week47/html/week47-solarized.html index 19684e17b..96a055e4b 100644 --- a/doc/pub/week47/html/week47-solarized.html +++ b/doc/pub/week47/html/week47-solarized.html @@ -81,7 +81,6 @@ div.toc p,a { None, 'first-attempt-at-a-minimization-approach'), ('Solving the equations', 2, None, 'solving-the-equations'), - ('Code Example', 2, None, 'code-example'), ('Problems with the Simpler Approach', 2, None, @@ -476,6 +475,7 @@ $$ \boldsymbol{x}^T\boldsymbol{w}+b=0. $$ +

    For figures, see handwritten notes for Thursday November 25.











    A \( p \)-dimensional space of features

    @@ -551,7 +551,9 @@ $$ f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0, $$ -

    as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.

    +

    as the function that determines the line \( L \) that separates two classes (our two features), see the figures in the handwritten notes for Thursday November 25. +. +

    Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).

    @@ -600,39 +602,13 @@ $$

    where \( \eta \) is our by now well-known learning rate.











    -

    Code Example

    +

    Problems with the Simpler Approach

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

    - -
    -
    -
    -
    -
    - 
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Problems with the Simpler Approach

    -

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

    @@ -777,6 +753,7 @@ $$









    Setting up the Problem

    +

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

    $$ {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], diff --git a/doc/pub/week47/html/week47.html b/doc/pub/week47/html/week47.html index 79418794d..7369ceef8 100644 --- a/doc/pub/week47/html/week47.html +++ b/doc/pub/week47/html/week47.html @@ -158,7 +158,6 @@ div.toc p,a { None, 'first-attempt-at-a-minimization-approach'), ('Solving the equations', 2, None, 'solving-the-equations'), - ('Code Example', 2, None, 'code-example'), ('Problems with the Simpler Approach', 2, None, @@ -553,6 +552,7 @@ $$ \boldsymbol{x}^T\boldsymbol{w}+b=0. $$ +

    For figures, see handwritten notes for Thursday November 25.











    A \( p \)-dimensional space of features

    @@ -628,7 +628,9 @@ $$ f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0, $$ -

    as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.

    +

    as the function that determines the line \( L \) that separates two classes (our two features), see the figures in the handwritten notes for Thursday November 25. +. +

    Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).

    @@ -677,39 +679,13 @@ $$

    where \( \eta \) is our by now well-known learning rate.











    -

    Code Example

    +

    Problems with the Simpler Approach

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

    - -
    -
    -
    -
    -
    - 
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - - -









    -

    Problems with the Simpler Approach

    -

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

    @@ -854,6 +830,7 @@ $$









    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/week47/ipynb/ipynb-week47-src.tar.gz b/doc/pub/week47/ipynb/ipynb-week47-src.tar.gz index f5c93d0b9bb2b68641026d8eb635a6a70a11ee94..9554402be0a2baab521222edb3c89d7c4bf118d9 100644 GIT binary patch delta 53 zcmX@M*yz||BR2VN4u%kwMz&Tq##T0_RyO8VHkMX4)>by42zx6VM=Kj=D;rlU8~0W= Ho&ysBf4>bu delta 53 zcmX@M*yz||BR2VN4u+pXjcl!KjIC@;t!&J#Y%Hy8tgUQ75%yL#j#f6#RyM9yHtwx# IJO?HM0GIC#)Bpeg diff --git a/doc/pub/week47/ipynb/week47.ipynb b/doc/pub/week47/ipynb/week47.ipynb index ac896dc25..514d3775d 100644 --- a/doc/pub/week47/ipynb/week47.ipynb +++ b/doc/pub/week47/ipynb/week47.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "11eedad1", + "id": "441a43d3", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "d229ce51", + "id": "9b7b8016", "metadata": { "editable": true }, @@ -29,7 +29,7 @@ }, { "cell_type": "markdown", - "id": "05269c87", + "id": "8f765cce", "metadata": { "editable": true }, @@ -52,7 +52,7 @@ }, { "cell_type": "markdown", - "id": "835c09a2", + "id": "d4cefcfd", "metadata": { "editable": true }, @@ -86,7 +86,7 @@ }, { "cell_type": "markdown", - "id": "873bd806", + "id": "e4693764", "metadata": { "editable": true }, @@ -108,7 +108,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "70acd877", + "id": "cd2ae27b", "metadata": { "collapsed": false, "editable": true @@ -187,7 +187,7 @@ }, { "cell_type": "markdown", - "id": "037c53af", + "id": "224d0cee", "metadata": { "editable": true }, @@ -207,7 +207,7 @@ }, { "cell_type": "markdown", - "id": "6937fa1a", + "id": "c383589e", "metadata": { "editable": true }, @@ -219,7 +219,7 @@ }, { "cell_type": "markdown", - "id": "5a04f85f", + "id": "3c8ea551", "metadata": { "editable": true }, @@ -232,7 +232,7 @@ }, { "cell_type": "markdown", - "id": "6b532ca6", + "id": "4a7aa762", "metadata": { "editable": true }, @@ -244,7 +244,17 @@ }, { "cell_type": "markdown", - "id": "7fd0bcd8", + "id": "b6eec1b9", + "metadata": { + "editable": true + }, + "source": [ + "For figures, see [handwritten notes](https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021) for Thursday November 25." + ] + }, + { + "cell_type": "markdown", + "id": "7ddfc2a4", "metadata": { "editable": true }, @@ -257,7 +267,7 @@ }, { "cell_type": "markdown", - "id": "60502389", + "id": "c1a7c663", "metadata": { "editable": true }, @@ -269,7 +279,7 @@ }, { "cell_type": "markdown", - "id": "cbe32b34", + "id": "6e56a6bd", "metadata": { "editable": true }, @@ -281,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "12186d6e", + "id": "5a75b5b8", "metadata": { "editable": true }, @@ -293,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "fcb8d56c", + "id": "3136f844", "metadata": { "editable": true }, @@ -303,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "adcc6c78", + "id": "d99fe2c5", "metadata": { "editable": true }, @@ -315,7 +325,7 @@ }, { "cell_type": "markdown", - "id": "ed8a91a8", + "id": "bfa14a88", "metadata": { "editable": true }, @@ -326,7 +336,7 @@ }, { "cell_type": "markdown", - "id": "d8ee2f04", + "id": "9eeec720", "metadata": { "editable": true }, @@ -338,7 +348,7 @@ }, { "cell_type": "markdown", - "id": "8a5282be", + "id": "7681b46e", "metadata": { "editable": true }, @@ -351,7 +361,7 @@ }, { "cell_type": "markdown", - "id": "6864075d", + "id": "bf112257", "metadata": { "editable": true }, @@ -363,7 +373,7 @@ }, { "cell_type": "markdown", - "id": "b8d30ca8", + "id": "df7a0279", "metadata": { "editable": true }, @@ -373,7 +383,7 @@ }, { "cell_type": "markdown", - "id": "a2d08a2d", + "id": "48d64b2e", "metadata": { "editable": true }, @@ -402,7 +412,7 @@ }, { "cell_type": "markdown", - "id": "66f55738", + "id": "657dd547", "metadata": { "editable": true }, @@ -414,7 +424,7 @@ }, { "cell_type": "markdown", - "id": "325b47fb", + "id": "635493a3", "metadata": { "editable": true }, @@ -426,12 +436,13 @@ }, { "cell_type": "markdown", - "id": "7d7fbf12", + "id": "942246f0", "metadata": { "editable": true }, "source": [ - "as the function that determines the line $L$ that separates two classes (our two features), see the figure here. \n", + "as the function that determines the line $L$ that separates two classes (our two features), see the figures in the [handwritten notes](https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021) for Thursday November 25.\n", + ". \n", "\n", "Any point defined by $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_2$ on the line $L$ will satisfy $\\boldsymbol{w}^T(\\boldsymbol{x}_1-\\boldsymbol{x}_2)=0$. \n", "\n", @@ -440,7 +451,7 @@ }, { "cell_type": "markdown", - "id": "91066108", + "id": "e531a14e", "metadata": { "editable": true }, @@ -452,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "3c5f028b", + "id": "6e49cdbc", "metadata": { "editable": true }, @@ -466,7 +477,7 @@ }, { "cell_type": "markdown", - "id": "01aea9cd", + "id": "a191b445", "metadata": { "editable": true }, @@ -478,7 +489,7 @@ }, { "cell_type": "markdown", - "id": "def8f1fc", + "id": "56fe3b52", "metadata": { "editable": true }, @@ -488,7 +499,7 @@ }, { "cell_type": "markdown", - "id": "034d7e72", + "id": "ef48a94c", "metadata": { "editable": true }, @@ -500,7 +511,7 @@ }, { "cell_type": "markdown", - "id": "a18e2f52", + "id": "f6fbcbae", "metadata": { "editable": true }, @@ -510,7 +521,7 @@ }, { "cell_type": "markdown", - "id": "27e7a61a", + "id": "b29775a4", "metadata": { "editable": true }, @@ -522,7 +533,7 @@ }, { "cell_type": "markdown", - "id": "e8188a27", + "id": "61d25a02", "metadata": { "editable": true }, @@ -534,7 +545,7 @@ }, { "cell_type": "markdown", - "id": "e7df215c", + "id": "88ad70f0", "metadata": { "editable": true }, @@ -546,7 +557,7 @@ }, { "cell_type": "markdown", - "id": "aef531b4", + "id": "8a9805b1", "metadata": { "editable": true }, @@ -556,7 +567,7 @@ }, { "cell_type": "markdown", - "id": "b73a9613", + "id": "38184889", "metadata": { "editable": true }, @@ -568,7 +579,7 @@ }, { "cell_type": "markdown", - "id": "93b93705", + "id": "68e207ee", "metadata": { "editable": true }, @@ -578,27 +589,17 @@ }, { "cell_type": "markdown", - "id": "2acb6d5c", - "metadata": { - "editable": true - }, - "source": [ - "## Code Example\n", - "\n", - "The equations we discussed above can be coded rather easily (the\n", - "framework is similar to what we developed for logistic\n", - "regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way." - ] - }, - { - "cell_type": "markdown", - "id": "42451c9c", + "id": "fe1446d6", "metadata": { "editable": true }, "source": [ "## Problems with the Simpler Approach\n", "\n", + "The equations we discussed above can be coded rather easily (the\n", + "framework is similar to what we developed for logistic\n", + "regression). \n", + "\n", "There are however problems with this approach, although it looks\n", "pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.\n", "\n", @@ -611,7 +612,7 @@ }, { "cell_type": "markdown", - "id": "875c0666", + "id": "14060d30", "metadata": { "editable": true }, @@ -627,7 +628,7 @@ }, { "cell_type": "markdown", - "id": "1815e5a6", + "id": "0fcf7b2d", "metadata": { "editable": true }, @@ -639,7 +640,7 @@ }, { "cell_type": "markdown", - "id": "e9e0cf87", + "id": "c6809e9b", "metadata": { "editable": true }, @@ -651,7 +652,7 @@ }, { "cell_type": "markdown", - "id": "7d4147d6", + "id": "915107ff", "metadata": { "editable": true }, @@ -663,7 +664,7 @@ }, { "cell_type": "markdown", - "id": "aa538edd", + "id": "148b8fbc", "metadata": { "editable": true }, @@ -673,7 +674,7 @@ }, { "cell_type": "markdown", - "id": "515cabfa", + "id": "20fe90e0", "metadata": { "editable": true }, @@ -685,7 +686,7 @@ }, { "cell_type": "markdown", - "id": "98ad7d18", + "id": "1f7f5ad4", "metadata": { "editable": true }, @@ -696,7 +697,7 @@ }, { "cell_type": "markdown", - "id": "1e1f5365", + "id": "405e59ec", "metadata": { "editable": true }, @@ -708,7 +709,7 @@ }, { "cell_type": "markdown", - "id": "d8b4a59e", + "id": "7c051028", "metadata": { "editable": true }, @@ -721,7 +722,7 @@ }, { "cell_type": "markdown", - "id": "110cb7ee", + "id": "07fbb08f", "metadata": { "editable": true }, @@ -734,7 +735,7 @@ }, { "cell_type": "markdown", - "id": "b769d022", + "id": "ca5e55e8", "metadata": { "editable": true }, @@ -746,7 +747,7 @@ }, { "cell_type": "markdown", - "id": "514d49d9", + "id": "24b89676", "metadata": { "editable": true }, @@ -756,7 +757,7 @@ }, { "cell_type": "markdown", - "id": "ce2e8468", + "id": "4b4db4f7", "metadata": { "editable": true }, @@ -768,7 +769,7 @@ }, { "cell_type": "markdown", - "id": "faae74f3", + "id": "2399f9c4", "metadata": { "editable": true }, @@ -778,7 +779,7 @@ }, { "cell_type": "markdown", - "id": "596f292e", + "id": "b7f7666e", "metadata": { "editable": true }, @@ -790,7 +791,7 @@ }, { "cell_type": "markdown", - "id": "50946c25", + "id": "76ae2593", "metadata": { "editable": true }, @@ -807,7 +808,7 @@ }, { "cell_type": "markdown", - "id": "c9803b27", + "id": "a415d18a", "metadata": { "editable": true }, @@ -819,7 +820,7 @@ }, { "cell_type": "markdown", - "id": "61dd7d63", + "id": "a7311b9b", "metadata": { "editable": true }, @@ -829,7 +830,7 @@ }, { "cell_type": "markdown", - "id": "f8371778", + "id": "c3370b3e", "metadata": { "editable": true }, @@ -841,7 +842,7 @@ }, { "cell_type": "markdown", - "id": "c9eaf0be", + "id": "ef69a59d", "metadata": { "editable": true }, @@ -851,7 +852,7 @@ }, { "cell_type": "markdown", - "id": "b829b68d", + "id": "32b0d248", "metadata": { "editable": true }, @@ -863,7 +864,7 @@ }, { "cell_type": "markdown", - "id": "ce089ead", + "id": "efce8b1d", "metadata": { "editable": true }, @@ -876,7 +877,7 @@ }, { "cell_type": "markdown", - "id": "ad85648b", + "id": "94c464d3", "metadata": { "editable": true }, @@ -888,7 +889,7 @@ }, { "cell_type": "markdown", - "id": "b2823efd", + "id": "330899ae", "metadata": { "editable": true }, @@ -900,7 +901,7 @@ }, { "cell_type": "markdown", - "id": "61eb6a55", + "id": "72b6b3c5", "metadata": { "editable": true }, @@ -910,7 +911,7 @@ }, { "cell_type": "markdown", - "id": "e3b50194", + "id": "e409858a", "metadata": { "editable": true }, @@ -924,7 +925,7 @@ }, { "cell_type": "markdown", - "id": "10fe759e", + "id": "6b3c4204", "metadata": { "editable": true }, @@ -934,7 +935,7 @@ }, { "cell_type": "markdown", - "id": "fa95693e", + "id": "94db4a8c", "metadata": { "editable": true }, @@ -946,7 +947,7 @@ }, { "cell_type": "markdown", - "id": "33dc27e7", + "id": "1799a58d", "metadata": { "editable": true }, @@ -956,7 +957,7 @@ }, { "cell_type": "markdown", - "id": "8a97e646", + "id": "04ee0d2d", "metadata": { "editable": true }, @@ -968,7 +969,7 @@ }, { "cell_type": "markdown", - "id": "9833d58a", + "id": "947f9c04", "metadata": { "editable": true }, @@ -978,7 +979,7 @@ }, { "cell_type": "markdown", - "id": "56748148", + "id": "ecf773f9", "metadata": { "editable": true }, @@ -990,7 +991,7 @@ }, { "cell_type": "markdown", - "id": "5aafb764", + "id": "bc8f11e0", "metadata": { "editable": true }, @@ -1004,7 +1005,7 @@ }, { "cell_type": "markdown", - "id": "82cd5c73", + "id": "1f6db8fe", "metadata": { "editable": true }, @@ -1016,18 +1017,19 @@ }, { "cell_type": "markdown", - "id": "35dc5cd9", + "id": "7ae6fca4", "metadata": { "editable": true }, "source": [ "## Setting up the Problem\n", + "\n", "In order to solve the above problem, we define the following Lagrangian function to be minimized" ] }, { "cell_type": "markdown", - "id": "2770972c", + "id": "7c8906c8", "metadata": { "editable": true }, @@ -1039,7 +1041,7 @@ }, { "cell_type": "markdown", - "id": "7d7c478c", + "id": "5b5adbd5", "metadata": { "editable": true }, @@ -1051,7 +1053,7 @@ }, { "cell_type": "markdown", - "id": "394e17a9", + "id": "f2c09c3f", "metadata": { "editable": true }, @@ -1063,7 +1065,7 @@ }, { "cell_type": "markdown", - "id": "6d085f58", + "id": "a3db7c50", "metadata": { "editable": true }, @@ -1073,7 +1075,7 @@ }, { "cell_type": "markdown", - "id": "5512ef2d", + "id": "ec5c7315", "metadata": { "editable": true }, @@ -1085,7 +1087,7 @@ }, { "cell_type": "markdown", - "id": "022d5960", + "id": "54afc174", "metadata": { "editable": true }, @@ -1095,7 +1097,7 @@ }, { "cell_type": "markdown", - "id": "3e32ab66", + "id": "42358128", "metadata": { "editable": true }, @@ -1107,7 +1109,7 @@ }, { "cell_type": "markdown", - "id": "1e4819da", + "id": "a1629247", "metadata": { "editable": true }, @@ -1118,7 +1120,7 @@ }, { "cell_type": "markdown", - "id": "d747b0ba", + "id": "4b650985", "metadata": { "editable": true }, @@ -1130,7 +1132,7 @@ }, { "cell_type": "markdown", - "id": "a6a4d86a", + "id": "3ca241de", "metadata": { "editable": true }, @@ -1144,7 +1146,7 @@ }, { "cell_type": "markdown", - "id": "28f83c49", + "id": "f8db86f9", "metadata": { "editable": true }, @@ -1156,7 +1158,7 @@ }, { "cell_type": "markdown", - "id": "582c0490", + "id": "7aabb47c", "metadata": { "editable": true }, @@ -1168,7 +1170,7 @@ }, { "cell_type": "markdown", - "id": "d91f1fba", + "id": "c8d61881", "metadata": { "editable": true }, @@ -1178,7 +1180,7 @@ }, { "cell_type": "markdown", - "id": "d8489e22", + "id": "11b7dc90", "metadata": { "editable": true }, @@ -1195,7 +1197,7 @@ }, { "cell_type": "markdown", - "id": "15cf6b01", + "id": "031eb330", "metadata": { "editable": true }, @@ -1206,7 +1208,7 @@ }, { "cell_type": "markdown", - "id": "7ae516c8", + "id": "36d7d3e9", "metadata": { "editable": true }, @@ -1219,7 +1221,7 @@ }, { "cell_type": "markdown", - "id": "a4475f57", + "id": "672614da", "metadata": { "editable": true }, @@ -1231,7 +1233,7 @@ }, { "cell_type": "markdown", - "id": "db157b5b", + "id": "70e17308", "metadata": { "editable": true }, @@ -1241,7 +1243,7 @@ }, { "cell_type": "markdown", - "id": "62e61a7b", + "id": "2aba9362", "metadata": { "editable": true }, @@ -1253,7 +1255,7 @@ }, { "cell_type": "markdown", - "id": "cc75cb92", + "id": "2dc26905", "metadata": { "editable": true }, @@ -1263,7 +1265,7 @@ }, { "cell_type": "markdown", - "id": "b28d44ef", + "id": "f4b79beb", "metadata": { "editable": true }, @@ -1275,7 +1277,7 @@ }, { "cell_type": "markdown", - "id": "5d9a5a73", + "id": "b52766fa", "metadata": { "editable": true }, @@ -1285,7 +1287,7 @@ }, { "cell_type": "markdown", - "id": "7014e7a3", + "id": "4cbe5c72", "metadata": { "editable": true }, @@ -1297,7 +1299,7 @@ }, { "cell_type": "markdown", - "id": "974813b7", + "id": "3fa36d56", "metadata": { "editable": true }, @@ -1307,7 +1309,7 @@ }, { "cell_type": "markdown", - "id": "d1691443", + "id": "b2f2f797", "metadata": { "editable": true }, @@ -1319,7 +1321,7 @@ }, { "cell_type": "markdown", - "id": "630ea9df", + "id": "924792c0", "metadata": { "editable": true }, @@ -1329,7 +1331,7 @@ }, { "cell_type": "markdown", - "id": "873d9437", + "id": "97545484", "metadata": { "editable": true }, @@ -1349,7 +1351,7 @@ }, { "cell_type": "markdown", - "id": "7479ef27", + "id": "d3b1f4d6", "metadata": { "editable": true }, @@ -1361,7 +1363,7 @@ }, { "cell_type": "markdown", - "id": "9d24f832", + "id": "0f6c0fd4", "metadata": { "editable": true }, @@ -1371,7 +1373,7 @@ }, { "cell_type": "markdown", - "id": "f54f1925", + "id": "7106ab0c", "metadata": { "editable": true }, @@ -1383,7 +1385,7 @@ }, { "cell_type": "markdown", - "id": "1ab5f910", + "id": "4be246c4", "metadata": { "editable": true }, @@ -1399,7 +1401,7 @@ }, { "cell_type": "markdown", - "id": "a8cfa315", + "id": "1a10e09c", "metadata": { "editable": true }, @@ -1411,7 +1413,7 @@ }, { "cell_type": "markdown", - "id": "928c273c", + "id": "5a6591cb", "metadata": { "editable": true }, @@ -1423,7 +1425,7 @@ }, { "cell_type": "markdown", - "id": "89c3a2a8", + "id": "0b6a60bb", "metadata": { "editable": true }, @@ -1433,7 +1435,7 @@ }, { "cell_type": "markdown", - "id": "3444a4f1", + "id": "9002c900", "metadata": { "editable": true }, @@ -1445,7 +1447,7 @@ }, { "cell_type": "markdown", - "id": "1c648831", + "id": "efaeee24", "metadata": { "editable": true }, @@ -1457,7 +1459,7 @@ }, { "cell_type": "markdown", - "id": "9a53ac4a", + "id": "94a0a96a", "metadata": { "editable": true }, @@ -1469,7 +1471,7 @@ }, { "cell_type": "markdown", - "id": "d93cd756", + "id": "35da28cd", "metadata": { "editable": true }, @@ -1479,7 +1481,7 @@ }, { "cell_type": "markdown", - "id": "41e268fc", + "id": "3e302d5a", "metadata": { "editable": true }, @@ -1491,7 +1493,7 @@ }, { "cell_type": "markdown", - "id": "fcbaafba", + "id": "efe155f7", "metadata": { "editable": true }, @@ -1501,7 +1503,7 @@ }, { "cell_type": "markdown", - "id": "3128acc6", + "id": "ff1f016f", "metadata": { "editable": true }, @@ -1513,7 +1515,7 @@ }, { "cell_type": "markdown", - "id": "68eee6ce", + "id": "59cab632", "metadata": { "editable": true }, @@ -1523,7 +1525,7 @@ }, { "cell_type": "markdown", - "id": "f68dee1a", + "id": "86e2728b", "metadata": { "editable": true }, @@ -1535,7 +1537,7 @@ }, { "cell_type": "markdown", - "id": "6aadb5c2", + "id": "fa9dd640", "metadata": { "editable": true }, @@ -1546,7 +1548,7 @@ }, { "cell_type": "markdown", - "id": "a8c28ca0", + "id": "8fb8d57e", "metadata": { "editable": true }, @@ -1558,7 +1560,7 @@ }, { "cell_type": "markdown", - "id": "b5a714d4", + "id": "e2726fd1", "metadata": { "editable": true }, @@ -1570,7 +1572,7 @@ }, { "cell_type": "markdown", - "id": "54ab9331", + "id": "1619157b", "metadata": { "editable": true }, @@ -1580,7 +1582,7 @@ }, { "cell_type": "markdown", - "id": "5b16a8d8", + "id": "3f7a7063", "metadata": { "editable": true }, @@ -1592,7 +1594,7 @@ }, { "cell_type": "markdown", - "id": "7e663d27", + "id": "035b199e", "metadata": { "editable": true }, @@ -1618,7 +1620,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "c37b6b57", + "id": "2564e935", "metadata": { "collapsed": false, "editable": true @@ -1675,7 +1677,7 @@ }, { "cell_type": "markdown", - "id": "c02cfa09", + "id": "a0f7aa27", "metadata": { "editable": true }, @@ -1687,7 +1689,7 @@ }, { "cell_type": "markdown", - "id": "23ce8aa9", + "id": "1aa783a3", "metadata": { "editable": true }, @@ -1699,7 +1701,7 @@ }, { "cell_type": "markdown", - "id": "56d1e78f", + "id": "d803e017", "metadata": { "editable": true }, @@ -1709,7 +1711,7 @@ }, { "cell_type": "markdown", - "id": "bcaa6bfb", + "id": "aabaaf68", "metadata": { "editable": true }, @@ -1721,7 +1723,7 @@ }, { "cell_type": "markdown", - "id": "c6889b36", + "id": "4e674340", "metadata": { "editable": true }, @@ -1731,7 +1733,7 @@ }, { "cell_type": "markdown", - "id": "bbb242b8", + "id": "08423b9f", "metadata": { "editable": true }, @@ -1743,7 +1745,7 @@ }, { "cell_type": "markdown", - "id": "54c69269", + "id": "cfe3d3fa", "metadata": { "editable": true }, @@ -1754,7 +1756,7 @@ }, { "cell_type": "markdown", - "id": "198117f5", + "id": "0654390f", "metadata": { "editable": true }, @@ -1766,7 +1768,7 @@ }, { "cell_type": "markdown", - "id": "bb6a092d", + "id": "cfc69ba5", "metadata": { "editable": true }, @@ -1776,7 +1778,7 @@ }, { "cell_type": "markdown", - "id": "7c7963db", + "id": "4f953fd1", "metadata": { "editable": true }, @@ -1788,7 +1790,7 @@ }, { "cell_type": "markdown", - "id": "0bb4529c", + "id": "be4b01d0", "metadata": { "editable": true }, @@ -1806,7 +1808,7 @@ }, { "cell_type": "markdown", - "id": "d0860da4", + "id": "9d221f7a", "metadata": { "editable": true }, @@ -1817,7 +1819,7 @@ }, { "cell_type": "markdown", - "id": "02cbb83e", + "id": "f70d38be", "metadata": { "editable": true }, @@ -1829,7 +1831,7 @@ }, { "cell_type": "markdown", - "id": "21ab8e0a", + "id": "9501dd3a", "metadata": { "editable": true }, @@ -1839,7 +1841,7 @@ }, { "cell_type": "markdown", - "id": "0ccde505", + "id": "4c89e3d6", "metadata": { "editable": true }, @@ -1856,7 +1858,7 @@ }, { "cell_type": "markdown", - "id": "d325c73a", + "id": "f455aee7", "metadata": { "editable": true }, @@ -1870,7 +1872,7 @@ }, { "cell_type": "markdown", - "id": "14abe1e0", + "id": "57e5ae96", "metadata": { "editable": true }, @@ -1885,7 +1887,7 @@ }, { "cell_type": "markdown", - "id": "4aa701ba", + "id": "0b64948c", "metadata": { "editable": true }, @@ -1897,7 +1899,7 @@ }, { "cell_type": "markdown", - "id": "de58ef40", + "id": "cd1ff8c0", "metadata": { "editable": true }, @@ -1925,7 +1927,7 @@ }, { "cell_type": "markdown", - "id": "aa95c803", + "id": "d1cd71e4", "metadata": { "editable": true }, @@ -1937,7 +1939,7 @@ }, { "cell_type": "markdown", - "id": "e44cbcef", + "id": "f944a50f", "metadata": { "editable": true }, @@ -1952,7 +1954,7 @@ }, { "cell_type": "markdown", - "id": "7d15bfb0", + "id": "03656fab", "metadata": { "editable": true }, @@ -1963,7 +1965,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "4516246b", + "id": "51a487e3", "metadata": { "collapsed": false, "editable": true @@ -2162,7 +2164,7 @@ }, { "cell_type": "markdown", - "id": "70711864", + "id": "b806a2e1", "metadata": { "editable": true }, @@ -2174,7 +2176,7 @@ }, { "cell_type": "markdown", - "id": "8effd67e", + "id": "7804736e", "metadata": { "editable": true }, @@ -2189,7 +2191,7 @@ }, { "cell_type": "markdown", - "id": "d02e2818", + "id": "d0f71fc0", "metadata": { "editable": true }, @@ -2206,7 +2208,7 @@ }, { "cell_type": "markdown", - "id": "76c05a44", + "id": "0cf90126", "metadata": { "editable": true }, @@ -2227,7 +2229,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "80889c0c", + "id": "ae4eb7cf", "metadata": { "collapsed": false, "editable": true @@ -2240,7 +2242,7 @@ }, { "cell_type": "markdown", - "id": "cf7d92a4", + "id": "0b0262af", "metadata": { "editable": true }, @@ -2250,7 +2252,7 @@ }, { "cell_type": "markdown", - "id": "2ef9611b", + "id": "9a3d1bb1", "metadata": { "editable": true }, @@ -2262,7 +2264,7 @@ }, { "cell_type": "markdown", - "id": "3a00902a", + "id": "dfda25e9", "metadata": { "editable": true }, @@ -2277,7 +2279,7 @@ }, { "cell_type": "markdown", - "id": "1cb22c82", + "id": "fcefe1f8", "metadata": { "editable": true }, @@ -2287,7 +2289,7 @@ }, { "cell_type": "markdown", - "id": "ba290e45", + "id": "20377892", "metadata": { "editable": true }, @@ -2306,7 +2308,7 @@ }, { "cell_type": "markdown", - "id": "9b9e61f8", + "id": "23078a4a", "metadata": { "editable": true }, @@ -2316,7 +2318,7 @@ }, { "cell_type": "markdown", - "id": "bf906cdd", + "id": "9b02e601", "metadata": { "editable": true }, @@ -2328,7 +2330,7 @@ }, { "cell_type": "markdown", - "id": "de6f4dda", + "id": "cfd8fd0c", "metadata": { "editable": true }, @@ -2338,7 +2340,7 @@ }, { "cell_type": "markdown", - "id": "5a0cf8d8", + "id": "5cbdb0ee", "metadata": { "editable": true }, @@ -2350,7 +2352,7 @@ }, { "cell_type": "markdown", - "id": "e4d78d0d", + "id": "67492dc9", "metadata": { "editable": true }, @@ -2360,7 +2362,7 @@ }, { "cell_type": "markdown", - "id": "dcc18a57", + "id": "36a5ac5b", "metadata": { "editable": true }, @@ -2372,7 +2374,7 @@ }, { "cell_type": "markdown", - "id": "17ef78ff", + "id": "86207d6c", "metadata": { "editable": true }, @@ -2383,7 +2385,7 @@ }, { "cell_type": "markdown", - "id": "2a994afc", + "id": "4a1859e1", "metadata": { "editable": true }, @@ -2395,7 +2397,7 @@ }, { "cell_type": "markdown", - "id": "b59ed115", + "id": "a1ec9520", "metadata": { "editable": true }, @@ -2407,7 +2409,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "2b8129eb", + "id": "991724bc", "metadata": { "collapsed": false, "editable": true @@ -2431,7 +2433,7 @@ }, { "cell_type": "markdown", - "id": "d41b30f0", + "id": "91fee137", "metadata": { "editable": true }, @@ -2443,7 +2445,7 @@ }, { "cell_type": "markdown", - "id": "71194f96", + "id": "56617f67", "metadata": { "editable": true }, @@ -2460,7 +2462,7 @@ }, { "cell_type": "markdown", - "id": "13cb5251", + "id": "d07335c7", "metadata": { "editable": true }, @@ -2474,7 +2476,7 @@ }, { "cell_type": "markdown", - "id": "2b6d7d6e", + "id": "eeae8340", "metadata": { "editable": true }, @@ -2484,7 +2486,7 @@ }, { "cell_type": "markdown", - "id": "8931b59e", + "id": "4b276877", "metadata": { "editable": true }, @@ -2499,7 +2501,7 @@ }, { "cell_type": "markdown", - "id": "03727abd", + "id": "0ee162a3", "metadata": { "editable": true }, @@ -2515,7 +2517,7 @@ }, { "cell_type": "markdown", - "id": "f72ea138", + "id": "9f4d8dd7", "metadata": { "editable": true }, @@ -2540,7 +2542,7 @@ }, { "cell_type": "markdown", - "id": "e57c523f", + "id": "7ab47dbb", "metadata": { "editable": true }, @@ -2569,7 +2571,7 @@ }, { "cell_type": "markdown", - "id": "7490d694", + "id": "dba69bc1", "metadata": { "editable": true }, @@ -2585,7 +2587,7 @@ }, { "cell_type": "markdown", - "id": "4f7bb20b", + "id": "fb548338", "metadata": { "editable": true }, @@ -2610,7 +2612,7 @@ }, { "cell_type": "markdown", - "id": "b83826b7", + "id": "9a1e615b", "metadata": { "editable": true }, @@ -2657,7 +2659,7 @@ }, { "cell_type": "markdown", - "id": "134d371b", + "id": "94260cc2", "metadata": { "editable": true }, @@ -2693,7 +2695,7 @@ }, { "cell_type": "markdown", - "id": "e131c86a", + "id": "e6a30863", "metadata": { "editable": true }, @@ -2716,7 +2718,7 @@ }, { "cell_type": "markdown", - "id": "2f7b0955", + "id": "cca199f2", "metadata": { "editable": true }, @@ -2739,7 +2741,7 @@ }, { "cell_type": "markdown", - "id": "49fa4b7b", + "id": "f23bfec8", "metadata": { "editable": true }, @@ -2759,7 +2761,7 @@ }, { "cell_type": "markdown", - "id": "7d5c5a1c", + "id": "9f04f253", "metadata": { "editable": true }, @@ -2775,7 +2777,7 @@ }, { "cell_type": "markdown", - "id": "d556fdb2", + "id": "1fe7e57e", "metadata": { "editable": true }, @@ -2803,7 +2805,7 @@ }, { "cell_type": "markdown", - "id": "a4ba04de", + "id": "6f00ca9d", "metadata": { "editable": true }, @@ -2825,7 +2827,7 @@ }, { "cell_type": "markdown", - "id": "82576d56", + "id": "02275c60", "metadata": { "editable": true }, @@ -2850,7 +2852,7 @@ }, { "cell_type": "markdown", - "id": "3957ee87", + "id": "6bcb8e9f", "metadata": { "editable": true }, @@ -2868,7 +2870,7 @@ }, { "cell_type": "markdown", - "id": "8f43da22", + "id": "f397cbc9", "metadata": { "editable": true }, @@ -2896,7 +2898,7 @@ }, { "cell_type": "markdown", - "id": "e08a1bc0", + "id": "e27568c5", "metadata": { "editable": true }, @@ -2910,7 +2912,7 @@ }, { "cell_type": "markdown", - "id": "cf1b3d0d", + "id": "7c5dca33", "metadata": { "editable": true }, @@ -2936,7 +2938,7 @@ }, { "cell_type": "markdown", - "id": "8dca4061", + "id": "75ea06c4", "metadata": { "editable": true }, @@ -2965,7 +2967,7 @@ }, { "cell_type": "markdown", - "id": "b9f9edee", + "id": "973910ed", "metadata": { "editable": true }, @@ -2982,7 +2984,7 @@ }, { "cell_type": "markdown", - "id": "08640fb4", + "id": "e8a9939d", "metadata": { "editable": true }, @@ -3004,7 +3006,7 @@ }, { "cell_type": "markdown", - "id": "9557e3f5", + "id": "22ee217d", "metadata": { "editable": true }, @@ -3022,7 +3024,7 @@ }, { "cell_type": "markdown", - "id": "8a30742f", + "id": "1539fe90", "metadata": { "editable": true }, @@ -3045,7 +3047,7 @@ }, { "cell_type": "markdown", - "id": "0cea4abf", + "id": "fd5ea5f8", "metadata": { "editable": true }, @@ -3064,7 +3066,7 @@ }, { "cell_type": "markdown", - "id": "f13fb9f6", + "id": "e629ac9a", "metadata": { "editable": true }, @@ -3080,7 +3082,7 @@ }, { "cell_type": "markdown", - "id": "81d6be62", + "id": "b8a18a9c", "metadata": { "editable": true }, @@ -3095,7 +3097,7 @@ }, { "cell_type": "markdown", - "id": "c9da5a3f", + "id": "79cbd92e", "metadata": { "editable": true }, @@ -3119,7 +3121,7 @@ }, { "cell_type": "markdown", - "id": "a1ea8499", + "id": "b26ea45e", "metadata": { "editable": true }, @@ -3131,7 +3133,7 @@ }, { "cell_type": "markdown", - "id": "ff723dbd", + "id": "307b4b7b", "metadata": { "editable": true }, @@ -3149,7 +3151,7 @@ }, { "cell_type": "markdown", - "id": "84fe806f", + "id": "5c066128", "metadata": { "editable": true }, @@ -3159,7 +3161,7 @@ }, { "cell_type": "markdown", - "id": "29932151", + "id": "721ef62a", "metadata": { "editable": true }, @@ -3177,7 +3179,7 @@ }, { "cell_type": "markdown", - "id": "5e6fb888", + "id": "9a832728", "metadata": { "editable": true }, @@ -3187,7 +3189,7 @@ }, { "cell_type": "markdown", - "id": "1f48a948", + "id": "ee7d0eba", "metadata": { "editable": true }, @@ -3206,7 +3208,7 @@ }, { "cell_type": "markdown", - "id": "62302009", + "id": "243fef20", "metadata": { "editable": true }, @@ -3218,7 +3220,7 @@ }, { "cell_type": "markdown", - "id": "03ad1fab", + "id": "4858119d", "metadata": { "editable": true }, @@ -3232,7 +3234,7 @@ }, { "cell_type": "markdown", - "id": "36861f13", + "id": "926ef74b", "metadata": { "editable": true }, @@ -3247,7 +3249,7 @@ }, { "cell_type": "markdown", - "id": "3ed52b2d", + "id": "a57111e6", "metadata": { "editable": true }, @@ -3265,7 +3267,7 @@ }, { "cell_type": "markdown", - "id": "c28fbd39", + "id": "8d6b130d", "metadata": { "editable": true }, @@ -3279,7 +3281,7 @@ }, { "cell_type": "markdown", - "id": "5ab7e1fd", + "id": "6b861446", "metadata": { "editable": true }, @@ -3297,7 +3299,7 @@ }, { "cell_type": "markdown", - "id": "deaa82f8", + "id": "01055af3", "metadata": { "editable": true }, @@ -3321,7 +3323,7 @@ }, { "cell_type": "markdown", - "id": "4a3b2c29", + "id": "a08e71c6", "metadata": { "editable": true }, @@ -3359,7 +3361,7 @@ }, { "cell_type": "markdown", - "id": "2b0f5287", + "id": "26ca4343", "metadata": { "editable": true }, @@ -3382,7 +3384,7 @@ }, { "cell_type": "markdown", - "id": "03fdfcaf", + "id": "d73601ed", "metadata": { "editable": true }, @@ -3418,7 +3420,7 @@ }, { "cell_type": "markdown", - "id": "230473cf", + "id": "224c0bc8", "metadata": { "editable": true }, @@ -3439,7 +3441,7 @@ }, { "cell_type": "markdown", - "id": "846d275b", + "id": "6e099fd1", "metadata": { "editable": true }, @@ -3461,7 +3463,7 @@ }, { "cell_type": "markdown", - "id": "ed7f7eed", + "id": "d6d3e71c", "metadata": { "editable": true }, @@ -3481,7 +3483,7 @@ }, { "cell_type": "markdown", - "id": "94f27695", + "id": "a5b322df", "metadata": { "editable": true }, @@ -3496,7 +3498,7 @@ }, { "cell_type": "markdown", - "id": "2ae2b5d9", + "id": "7bc64edb", "metadata": { "editable": true }, @@ -3514,7 +3516,7 @@ }, { "cell_type": "markdown", - "id": "c8170bd4", + "id": "636cbc0f", "metadata": { "editable": true }, @@ -3544,7 +3546,7 @@ }, { "cell_type": "markdown", - "id": "770c9f3b", + "id": "57c30225", "metadata": { "editable": true }, @@ -3573,7 +3575,7 @@ }, { "cell_type": "markdown", - "id": "24ab7dc9", + "id": "c51bbe72", "metadata": { "editable": true }, @@ -3604,7 +3606,7 @@ }, { "cell_type": "markdown", - "id": "daff58eb", + "id": "d8561e2a", "metadata": { "editable": true }, @@ -3627,7 +3629,7 @@ }, { "cell_type": "markdown", - "id": "49afc03f", + "id": "a2c6d822", "metadata": { "editable": true }, @@ -3645,7 +3647,7 @@ }, { "cell_type": "markdown", - "id": "6279873b", + "id": "01529986", "metadata": { "editable": true }, @@ -3668,7 +3670,7 @@ }, { "cell_type": "markdown", - "id": "10760689", + "id": "6fc3228e", "metadata": { "editable": true }, @@ -3689,7 +3691,7 @@ }, { "cell_type": "markdown", - "id": "5245b378", + "id": "a7857509", "metadata": { "editable": true }, @@ -3705,7 +3707,7 @@ }, { "cell_type": "markdown", - "id": "7bce2b05", + "id": "595ec8ca", "metadata": { "editable": true }, diff --git a/doc/src/week47/week47.do.txt b/doc/src/week47/week47.do.txt index 6ded92c4b..34554d2d0 100644 --- a/doc/src/week47/week47.do.txt +++ b/doc/src/week47/week47.do.txt @@ -162,6 +162,8 @@ We can then rewrite the above equation as \] !et +For figures, see "handwritten notes":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021" for Thursday November 25. + !split ===== A $p$-dimensional space of features ===== @@ -236,7 +238,8 @@ Let us define the function f(x) = \bm{w}^T\bm{x}+b = 0, \] !et -as the function that determines the line $L$ that separates two classes (our two features), see the figure here. +as the function that determines the line $L$ that separates two classes (our two features), see the figures in the "handwritten notes":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021" for Thursday November 25. +. Any point defined by $\bm{x}_i$ and $\bm{x}_2$ on the line $L$ will satisfy $\bm{w}^T(\bm{x}_1-\bm{x}_2)=0$. @@ -292,19 +295,15 @@ and where $\eta$ is our by now well-known learning rate. -!split -===== Code Example ===== -The equations we discussed above can be coded rather easily (the -framework is similar to what we developed for logistic -regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way. -!bc pycod - -!ec !split ===== Problems with the Simpler Approach ===== +The equations we discussed above can be coded rather easily (the +framework is similar to what we developed for logistic +regression). + There are however problems with this approach, although it looks pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes. @@ -458,6 +457,7 @@ If we have a set of constraints $\phi_k$ we have the equations !split ===== Setting up the Problem ===== + In order to solve the above problem, we define the following Lagrangian function to be minimized !bt \[