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By
Alexandre Kowalczyk
Foreword by Daniel Jebaraj
3
Copyright © 2017 by Syncfusion, Inc.
2501 Aerial Center Parkway
Suite 200
Morrisville, NC 27560
USA
All rights reserved.
Important licensing information. Please read.
This book is available for free download from www.syncfusion.com on completion of a
registration form.
If you obtained this book from any other source, please register and download a free copy from
www.syncfusion.com.
This book is licensed for reading only if obtained from www.syncfusion.com.
This book is licensed strictly for personal or educational use.
Redistribution in any form is prohibited.
The authors and copyright holders provide absolutely no warranty for any information provided.
The authors and copyright holders shall not be liable for any claim, damages, or any other
liability arising from, out of, or in connection with the information in this book.
Please do not use this book if the listed terms are unacceptable.
Use shall constitute acceptance of the terms listed.
SYNCFUSION, SUCCINCTLY, DELIVER INNOVATION WITH EASE, ESSENTIAL, and .NET
ESSENTIALS are the registered trademarks of Syncfusion, Inc.
Technical Reviewer: James McCaffrey
Copy Editor: Courtney Wright
Acquisitions Coordinator: Hillary Bowling, online marketing manager, Syncfusion, Inc.
Proofreader: John Elderkin
4
Table of Contents
The Story behind the Succinctly Series of Books ................................................................. 8
About the Author ....................................................................................................................10
Preface .....................................................................................................................................11
Introduction .............................................................................................................................12
Chapter 1 Prerequisites .........................................................................................................13
Vectors .................................................................................................................................13
What is a vector? .............................................................................................................13
The dot product................................................................................................................17
Understanding linear separability .........................................................................................21
Linearly separable data ....................................................................................................21
Hyperplanes .........................................................................................................................24
What is a hyperplane? .....................................................................................................24
Understanding the hyperplane equation ..........................................................................25
Classifying data with a hyperplane ...................................................................................26
How can we find a hyperplane (separating the data or not)?............................................27
Summary ..............................................................................................................................28
Chapter 2 The Perceptron .....................................................................................................29
Presentation .........................................................................................................................29
The Perceptron learning algorithm........................................................................................29
Understanding the update rule .........................................................................................31
Convergence of the algorithm ..........................................................................................35
Understanding the limitations of the PLA .........................................................................35
Summary ..............................................................................................................................38
5
Chapter 3 The SVM Optimization Problem ...........................................................................39
SVMs search for the optimal hyperplane ..............................................................................39
How can we compare two hyperplanes? ..............................................................................39
Using the equation of the hyperplane ...............................................................................39
Problem with examples on the negative side ...................................................................41
Does the hyperplane correctly classify the data? .............................................................42
Scale invariance ..............................................................................................................43
What is an optimization problem? .........................................................................................48
Unconstrained optimization problem ................................................................................48
Constrained optimization problem ....................................................................................49
How do we solve an optimization problem? .....................................................................51
The SVMs optimization problem ...........................................................................................51
Summary ..............................................................................................................................53
Chapter 4 Solving the Optimization Problem .......................................................................54
Lagrange multipliers .............................................................................................................54
The method of Lagrange multipliers .................................................................................54
The SVM Lagrangian problem .........................................................................................54
The Wolfe dual problem .......................................................................................................55
Karush-Kuhn-Tucker conditions ...........................................................................................57
Stationarity condition .......................................................................................................58
Primal feasibility condition ................................................................................................58
Dual feasibility condition ..................................................................................................58
Complementary slackness condition ................................................................................59
What to do once we have the multipliers? ............................................................................59
Compute w ......................................................................................................................59
Compute b .......................................................................................................................59
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