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Learning with Kernels 1st Edition by Bernhard Schlkopf

Learning with Kernels 1st Edition by Bernhard Scholkopf, ISBN-13: 978-0262194754

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Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond 1st Edition by Bernhard Scholkopf, ISBN-13: 978-0262194754

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  • Publisher: ‎ The MIT Press; 1st edition (December 15, 2001)
  • Language: ‎ English
  • 644 pages
  • ISBN-10: ‎ 0262194759
  • ISBN-13: ‎ 978-0262194754

A comprehensive introduction to Support Vector Machines and related kernel methods.

In the 1990s, a new type of learning algorithm was developed, based on results from statistical learning theory: the Support Vector Machine (SVM). This gave rise to a new class of theoretically elegant learning machines that use a central concept of SVMs―-kernels―for a number of learning tasks. Kernel machines provide a modular framework that can be adapted to different tasks and domains by the choice of the kernel function and the base algorithm. They are replacing neural networks in a variety of fields, including engineering, information retrieval, and bioinformatics.

Learning with Kernels provides an introduction to SVMs and related kernel methods. Although the book begins with the basics, it also includes the latest research. It provides all of the concepts necessary to enable a reader equipped with some basic mathematical knowledge to enter the world of machine learning using theoretically well-founded yet easy-to-use kernel algorithms and to understand and apply the powerful algorithms that have been developed over the last few years.

Table of Contents:

Series Foreword xiii
Preface xv
1 A Tutorial Introduction 1
1.1 Data Representation and Similarity . . . . . . . . . . . . . . . . . . . 1
1.2 A Simple Pattern Recognition Algorithm . . . . . . . . . . . . . . . 4
1.3 Some Insights From Statistical Learning Theory . . . . . . . . . . . 6
1.4 Hyperplane Classifiers . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Support Vector Classification . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Support Vector Regression . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Kernel Principal Component Analysis . . . . . . . . . . . . . . . . . 19
1.8 Empirical Results and Implementations . . . . . . . . . . . . . . . . 21
I CONCEPTS AND TOOLS 23
2 Kernels 25
2.1 Product Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 The Representation of Similarities in Linear Spaces . . . . . . . . . . 29
2.3 Examples and Properties of Kernels . . . . . . . . . . . . . . . . . . 45
2.4 The Representation of Dissimilarities in Linear Spaces . . . . . . . . 48
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Risk and Loss Functions 61
3.1 Loss Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 Test Error and Expected Risk . . . . . . . . . . . . . . . . . . . . . . 65
3.3 A Statistical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Robust Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 Regularization 87
4.1 The Regularized Risk Functional . . . . . . . . . . . . . . . . . . . . 88
4.2 The Representer Theorem . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 Regularization Operators . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Translation Invariant Kernels . . . . . . . . . . . . . . . . . . . . . . 96
4.5 Translation Invariant Kernels in Higher Dimensions . . . . . . . . . 105
4.6 Dot Product Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7 Multi-Output Regularization . . . . . . . . . . . . . . . . . . . . . . 113
4.8 Semiparametric Regularization . . . . . . . . . . . . . . . . . . . . . 115
4.9 Coefficient Based Regularization . . . . . . . . . . . . . . . . . . . . 118
4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5 Elements of Statistical Learning Theory 125
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 When Does LearningWork: the Question of Consistency . . . . . . 131
5.4 Uniform Convergence and Consistency . . . . . . . . . . . . . . . . 131
5.5 How to Derive a VC Bound . . . . . . . . . . . . . . . . . . . . . . . 134
5.6 A Model Selection Example . . . . . . . . . . . . . . . . . . . . . . . 144
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Optimization 149
6.1 Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.2 Unconstrained Problems . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3 Constrained Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.4 Interior Point Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.5 Maximum Search Problems . . . . . . . . . . . . . . . . . . . . . . . 179
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
II SUPPORT VECTOR MACHINES 187
7 Pattern Recognition 189
7.1 Separating Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.2 The Role of the Margin . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.3 Optimal Margin Hyperplanes . . . . . . . . . . . . . . . . . . . . . . 196
7.4 Nonlinear Support Vector Classifiers . . . . . . . . . . . . . . . . . . 200
7.5 Soft Margin Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 204
7.6 Multi-Class Classification . . . . . . . . . . . . . . . . . . . . . . . . 211
7.7 Variations on a Theme . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.8 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8 Single-Class Problems: Quantile Estimation and Novelty Detection 227
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.2 A Distribution’s Support and Quantiles . . . . . . . . . . . . . . . . 229
8.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.5 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8.7 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
9 Regression Estimation 251
9.1 Linear Regression with Insensitive Loss Function . . . . . . . . . . . 251
9.2 Dual Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.3 -SV Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
9.4 Convex Combinations and 1-Norms . . . . . . . . . . . . . . . . . . 266
9.5 Parametric Insensitivity Models . . . . . . . . . . . . . . . . . . . . . 269
9.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
9.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
10 Implementation 279
10.1 Tricks of the Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
10.2 Sparse Greedy Matrix Approximation . . . . . . . . . . . . . . . . . 288
10.3 Interior Point Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 295
10.4 Subset Selection Methods . . . . . . . . . . . . . . . . . . . . . . . . 300
10.5 Sequential Minimal Optimization . . . . . . . . . . . . . . . . . . . . 305
10.6 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
10.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
11 Incorporating Invariances 333
11.1 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
11.2 Transformation Invariance . . . . . . . . . . . . . . . . . . . . . . . . 335
11.3 The Virtual SV Method . . . . . . . . . . . . . . . . . . . . . . . . . . 337
11.4 Constructing Invariance Kernels . . . . . . . . . . . . . . . . . . . . 343
11.5 The Jittered SV Method . . . . . . . . . . . . . . . . . . . . . . . . . . 354
11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
11.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
12 Learning Theory Revisited 359
12.1 Concentration of Measure Inequalities . . . . . . . . . . . . . . . . . 360
12.2 Leave-One-Out Estimates . . . . . . . . . . . . . . . . . . . . . . . . 366
12.3 PAC-Bayesian Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 381
12.4 Operator-Theoretic Methods in Learning Theory . . . . . . . . . . . 391
12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
12.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
III KERNEL METHODS 405
13 Designing Kernels 407
13.1 Tricks for Constructing Kernels . . . . . . . . . . . . . . . . . . . . . 408
13.2 String Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
13.3 Locality-Improved Kernels . . . . . . . . . . . . . . . . . . . . . . . . 414
13.4 Natural Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
14 Kernel Feature Extraction 427
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
14.2 Kernel PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
14.3 Kernel PCA Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 437
14.4 A Framework for Feature Extraction . . . . . . . . . . . . . . . . . . 442
14.5 Algorithms for Sparse KFA . . . . . . . . . . . . . . . . . . . . . . . 447
14.6 KFA Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
14.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
14.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
15 Kernel Fisher Discriminant 457
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
15.2 Fisher’s Discriminant in Feature Space . . . . . . . . . . . . . . . . . 458
15.3 Efficient Training of Kernel Fisher Discriminants . . . . . . . . . . . 460
15.4 Probabilistic Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
15.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
15.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
16 Bayesian Kernel Methods 469
16.1 Bayesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
16.2 Inference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
16.3 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
16.4 Implementation of Gaussian Processes . . . . . . . . . . . . . . . . . 488
16.5 Laplacian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
16.6 Relevance Vector Machines . . . . . . . . . . . . . . . . . . . . . . . 506
16.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
16.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
17 Regularized Principal Manifolds 517
17.1 A Coding Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 518
17.2 A Regularized Quantization Functional . . . . . . . . . . . . . . . . 522
17.3 An Algorithm for Minimizing Rreg[ f] . . . . . . . . . . . . . . . . . 526
17.4 Connections to Other Algorithms . . . . . . . . . . . . . . . . . . . . 529
17.5 Uniform Convergence Bounds . . . . . . . . . . . . . . . . . . . . . 533
17.6 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
17.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
17.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
18 Pre-Images and Reduced Set Methods 543
18.1 The Pre-Image Problem . . . . . . . . . . . . . . . . . . . . . . . . . 544
18.2 Finding Approximate Pre-Images . . . . . . . . . . . . . . . . . . . . 547
18.3 Reduced Set Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
18.4 Reduced Set Selection Methods . . . . . . . . . . . . . . . . . . . . . 554
18.5 Reduced Set Construction Methods . . . . . . . . . . . . . . . . . . . 561
18.6 Sequential Evaluation of Reduced Set Expansions . . . . . . . . . . 564
18.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
18.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
A Addenda 569
A.1 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
A.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
B Mathematical Prerequisites 575
B.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
B.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
B.3 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
References 591
Index 617
Notation and Symbols 625

Bernhard Schölkopf is Director at the Max Planck Institute for Intelligent Systems in Tübingen, Germany. He is coauthor of Learning with Kernels (2002) and is a coeditor of Advances in Kernel Methods: Support Vector Learning (1998), Advances in Large-Margin Classifiers (2000), and Kernel Methods in Computational Biology (2004), all published by the MIT Press.

Alexander J. Smola is Senior Principal Researcher and Machine Learning Program Leader at National ICT Australia/Australian National University, Canberra.

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