Support Vector Machines

By: Ingo Steinwart, Andreas Christmann

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Hardcover | 12 August 2008

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This book explains the principles that make support vector machines (SVMs) a successful modelling and prediction tool for a variety of applications. The authors present the basic ideas of SVMs together with the latest developments and current research questions in a unified style. They identify three reasons for the success of SVMs: their ability to learn well with only a very small number of free parameters, their robustness against several types of model violations and outliers, and their computational efficiency compared to several other methods.

Since their appearance in the early nineties, support vector machines and related kernel-based methods have been successfully applied in diverse fields of application such as bioinformatics, fraud detection, construction of insurance tariffs, direct marketing, and data and text mining. As a consequence, SVMs now play an important role in statistical machine learning and are used not only by statisticians, mathematicians, and computer scientists, but also by engineers and data analysts.

The book provides a unique in-depth treatment of both fundamental and recent material on SVMs that so far has been scattered in the literature. The book can thus serve as both a basis for graduate courses and an introduction for statisticians, mathematicians, and computer scientists. It further provides a valuable reference for researchers working in the field.

The book covers all important topics concerning support vector machines such as: loss functions and their role in the learning process; reproducing kernel Hilbert spaces and their properties; a thorough statistical analysis that uses both traditional uniform bounds and more advanced localized techniques based on Rademacher averages and Talagrand's inequality; a detailed treatment of classification and regression; a detailed robustness analysis; and a description of some of the most recent implementation techniques. To make the book self-contained, an extensive appendix is added which provides the reader with the necessary background from statistics, probability theory, functional analysis, convex analysis, and topology.

Industry Reviews

From the reviews:

"This book has many remarkable qualities which make it commendable to a large mathematical audience. ...It is probably the first book on this topic...which is genuinely aimed at a mathematician reader. No technical issue is avoided, and fine points like measurability, integrability, existence and regularity of solutions, etc., are addressed with due rigor and precision. ...The authors take special care to make the book self-contained and accessible to non-specialists...always including very detailed proofs for all results. A substantial appendix acts as a handy reference of fundamental results of analysis and probability needed throughout the book, even including a full proof of Talagrand's concentration inequality. Many well-thought -out exercises very nicely complete each chapter. Finally, the book as a whole, though voluminous and presenting for the most part some very recent results, always stays very coherent to its choices and goals, and obviously a lot of effort has gone into a clear organization of the material. This work is bound to be recognized as a classic reference on this topic." (MathSciNet)

"This book presents an extensive account of ... Support Vector Machines (SVMs). ... The book has many remarkable qualities which make it commendable to a large mathematical audience. First of all it is probably the first book on this topic ... which is genuinely aimed at a mathematician reader. ... Secondly, the authors take special care to make the book self contained and accessible to non-specialists ... . Many well thought-out exercises very nicely complete each chapter. ... a classic reference on this topic." (Gilles Blanchard, Mathematical Reviews, Issue 2010 f)

"A mathematically elaborated topic of support vector machines in a book full with definitions and lemmas. It presents support vector machines (SVMs) as a successful modeling and prediction tool with different examples. This book has 12 chapters and 9 appendices that introduce also marginal applications of SVMs. ... This book is ... suitable as a textbook on SVMs for graduate courses ... ." (Adriana Hornikova, Technometrics, Vol. 53 (2), May, 2011)

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Paperback

Published: 31st October 2014

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Preface	p. vii
Reading Guide	p. xi
Introduction	p. 1
Statistical Learning	p. 1
Support Vector Machines: An Overview	p. 7
History of SVMs and Geometrical Interpretation	p. 13
Alternatives to SVMs	p. 19
Loss Functions and Their Risks	p. 21
Loss Functions: Definition and Examples	p. 21
Basic Properties of Loss Functions and Their Risks	p. 28
Margin-Based Losses for Classification Problems	p. 34
Distance-Based Losses for Regression Problems	p. 38
Further Reading and Advanced Topics	p. 45
Summary	p. 46
Exercises	p. 46
Surrogate Loss Functions (*)	p. 49
Inner Risks and the Calibration Function	p. 51
Asymptotic Theory of Surrogate Losses	p. 60
Inequalities between Excess Risks	p. 63
Surrogates for Unweighted Binary Classification	p. 71
Surrogates for Weighted Binary Classification	p. 76
Template Loss Functions	p. 80
Surrogate Losses for Regression Problems	p. 81
Surrogate Losses for the Density Level Problem	p. 93
Self-Calibrated Loss Functions	p. 97
Further Reading and Advanced Topics	p. 105
Summary	p. 106
Exercises	p. 107
Kernels and Reproducing Kernel Hilbert Spaces	p. 111
Basic Properties and Examples of Kernels	p. 112
The Reproducing Kernel Hilbert Space of a Kernel	p. 119
Properties of RKHSs	p. 124
Gaussian Kernels and Their RKHSs	p. 132
Mercer's Theorem (*)	p. 149
Large Reproducing Kernel Hilbert Spaces	p. 151
Further Reading and Advanced Topics	p. 159
Summary	p. 161
Exercises	p. 162
Infinite-Sample Versions of Support Vector Machines	p. 165
Existence and Uniqueness of SVM Solutions	p. 166
A General Representer Theorem	p. 169
Stability of Infinite-Sample SVMs	p. 173
Behavior for Small Regularization Parameters	p. 178
Approximation Error of RKHSs	p. 187
Further Reading and Advanced Topics	p. 197
Summary	p. 200
Exercises	p. 200
Basic Statistical Analysis of SVMs	p. 203
Notions of Statistical learning	p. 204
Basic Concentration Inequalities	p. 210
Statistical Analysis of Empirical Risk Minimization	p. 218
Basic Oracle Inequalities for SVMs	p. 223
Data-Dependent Parameter Selection for SVMs	p. 229
Further Reading and Advanced Topics	p. 234
Summary	p. 235
Exercises	p. 236
Advanced Statistical Analysis of SVMs (*)	p. 239
Why Do We Need a Refined Analysis?	p. 240
A Refined Oracle Inequality for ERM	p. 242
Some Advanced Machinery	p. 246
Refined Oracle Inequalities for SVMs	p. 258
Some Bounds on Average Entropy Numbers	p. 270
Further Reading and Advanced Topics	p. 279
Summary	p. 282
Exercises	p. 283
Support Vector Machines for Classification	p. 287
Basic Oracle Inequalities for Classifying with SVMs	p. 288
Classifying with SVMs Using Gaussian Kernels	p. 290
Advanced Concentration Results for SVMs (*)	p. 307
Sparseness of SVMs Using the Hinge Loss	p. 310
Classifying with other Margin-Based Losses (*)	p. 314
Further Reading and Advanced Topics	p. 326
Summary	p. 329
Exercises	p. 330
Support Vector Machines for Regression	p. 333
Introduction	p. 333
Consistency	p. 335
SVMs for Quantile Regression	p. 340
Numerical Results for Quantile Regression	p. 344
Median Regression with the eps-Insensitive Loss (*)	p. 348
Further Reading and Advanced Topics	p. 352
Summary	p. 353
Exercises	p. 353
Robustness	p. 355
Motivation	p. 356
Approaches to Robust Statistics	p. 362
Robustness of SVMs for Classification	p. 368
Robustness of SVMs for Regression (*)	p. 379
Robust Learning from Bites (*)	p. 391
Further Reading and Advanced Topics	p. 403
Summary	p. 408
Exercises	p. 409
Computational Aspects	p. 411
SVMs, Convex Programs, and Duality	p. 412
Implementation Techniques	p. 420
Determination of Hyperparameters	p. 443
Software Packages	p. 448
Further Reading and Advanced Topics	p. 450
Summary	p. 452
Exercises	p. 453
Data Mining	p. 455
Introduction	p. 456
CRISP-DM Strategy	p. 457
Role of SVMs in Data Mining	p. 467
Software Tools for Data Mining	p. 467
Further Reading and Advanced Topics	p. 468
Summary	p. 469
Exercises	p. 469
Appendix	p. 471
Basic Equations, Inequalities, and Functions	p. 471
Topology	p. 475
Measure and Integration Theory	p. 479
Some Basic Facts	p. 480
Measures on Topological Spaces	p. 486
Aumann's Measurable Selection Principle	p. 487
Probability Theory and Statistics	p. 489
Some Basic Facts	p. 489
Some Limit Theorems	p. 492
The Weak* Topology and Its Metrization	p. 494
Functional Analysis	p. 497
Essentials on Banach Spaces and Linear Operators	p. 497
Hilbert Spaces	p. 501
The Calculus in Normed Spaces	p. 507
Banach Space Valued Integration	p. 508
Some Important Banach Spaces	p. 511
Entropy Numbers	p. 516
Convex Analysis	p. 519
Basic Properties of Convex Functions	p. 520
Subdifferential Calculus for Convex Functions	p. 523
Some Further Notions of Convexity	p. 526
The Fenchel-Legendre Bi-conjugate	p. 529
Convex Programs and Lagrange Multipliers	p. 530
Complex Analysis	p. 534
Inequalities Involving Rademacher Sequences	p. 534
Talagrand's Inequality	p. 538
References	p. 553
Notation and Symbols	p. 579
Abbreviations	p. 583
Author Index	p. 585
Subject Index	p. 591
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Support Vector Machines

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