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Statistical learning theory (Loan 36 times)

Material type
단행본
Personal Author
Title Statement
Statistical learning theory / Vladimir N. Vapnik.
Publication, Distribution, etc
New York :   Wiley,   c1998.
Physical Medium
xxiv, 736 p. : ill. ; 25 cm.
Series Statement
Adaptive and learning systems for signal processing, communications, and control
ISBN
0471030031 (acid-free paper)
General Note
"A Wiley-Interscience publication."
Bibliography, Etc. Note
Includes bibliographical references (p. 723-732) and index.
Subject Added Entry-Topical Term
Computational learning theory.
 000 00998camuu2200277 a 4500 001 000000772623 005 20020618150140 008 970822s1998 nyua b 001 0 eng 010 ▼a 97037075 015 ▼a GB98-74066 020 ▼a 0471030031 (acid-free paper) 040 ▼a DLC ▼c DLC ▼d UKM ▼d 211009 049 1 ▼l 121063702 ▼f 과학 050 0 0 ▼a Q325.7 ▼b .V38 1998 082 0 0 ▼a 006.3/1 ▼2 21 090 ▼a 006.31 ▼b V286s 100 1 ▼a Vapnik, Vladimir Naumovich. 245 1 0 ▼a Statistical learning theory / ▼c Vladimir N. Vapnik. 260 ▼a New York : ▼b Wiley, ▼c c1998. 300 ▼a xxiv, 736 p. : ▼b ill. ; ▼c 25 cm. 440 0 ▼a Adaptive and learning systems for signal processing, communications, and control 500 ▼a "A Wiley-Interscience publication." 504 ▼a Includes bibliographical references (p. 723-732) and index. 650 0 ▼a Computational learning theory.

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Contents information

CONTENTS
PREFACE = xxi
Introduction : The Problem of Induction and Statistical Inference = 1
0.1 Learning Paradigm in Statistics = 1
0.2 Two Approaches to Statistical Inference : Particular (Parametirc Inference) and General (Noparametric Inference) = 2
0.3 The Paradigm Created by the Parametric Approach = 4
0.4 Shortcoming of the Parametric Paradigm = 5
0.5 After the Classical Paradigm = 6
0.6 The Renaissance = 7
0.7 The Generalization of the Glivento-Canteli-Kolmogorov Theory = 8
0.8 The Structural Risk Minimization Principle = 10
0.9 The Main Principle of Inference from a Small Sample Size = 11
0.10 What This Book is About = 13
1 THEORY OF LEARNING AND GENERALIZATION
1 Two Approaches to the Learning Problem = 19
1.1 General Model of Learning from Examples = 19
1.2 The Problem of Minimizing the Risk Function from Empirical Data = 21
1.3 The Problem of Pattern Recognition = 24
1.4 The Problem of Regression Estimation = 26
1.5 Problem of Interpreting Results of Indirect Measuring = 28
1.6 The Problem of Density Estimation (the Fisher-Wald Setting) = 30
1.7 Induction Principles for Minimizing the Risk Functional on the Basis of Empirical Data = 32
1.8 Classical Methods for Solving the Function Estimation Problems = 33
1.9 Inentification of Stochastic Objects : Estimation of the Densities and Conditional Densities = 35
1.10 The Problem of Solving an Approximatery Determined Integral Equation = 38
1.11 Glivenko-Cantelli Theorem = 39
1.11.1 Convergence in Probability and Almost Sure Convergence = 40
1.11.2 Glivenko-Cantelli Theorem = 42
1.11.3 Three Important Statistical Laws = 42
1.12 Ill-Posed Problems = 44
1.13 The Sturucture of the Learning Theory = 48
Appendix to Chapter 1 : Methods for Solving Ⅲ-Posed Problems
A1.1 The Problem of Solving an Operator Equation = 51
A1.2 Problems Well-Posed in Tikhonov's Sens = 53
A1.3 The Regularization Method = 54
A1.3.1 Idea of Regularization Method = 54
A1.3.2 Main Theorems About the Regularization Method = 55
2 Estimation of the Probability Measure and Problem of Learning
2.1 Probability Model of a Random Experiment = 59
2.2 The Basic Problem of Statistics = 61
2.2.1 The Basic Problems of Probability and Statistics = 61
2.2.2 Uniform Convergence of Probability Measure Estimates = 62
2.3 Conditions for the Uniform Convergence of Estimates to the Unknown Probability Measure = 65
2.3.1 Structure of Distribution Function = 65
2.3.2 Estimator that Provides Uniform Convergence = 68
2.4 Partial Uniform Convergence and Generalization of Glivenko-Cantelli Theorem = 69
2.4.1 Definition of Partial Uniform Convergence = 69
2.4.2 Generalization of the Glivenko-Cantelli Problem = 71
2.5 Minimizing the Risk Functional Under the Condition of Uniform Convergence of Probability Measure Estimates = 72
2.6 Minimizing the Risk Function Under the Condition of Partial Uniform Convergence of Probability Measure Estimates = 74
2.7 Remarks About Modes of Convergence of the Probability Measure Estimates and Statements of the Learning Problems = 77
3 Conditions for Consistency of Empirical Risk Minimization Principle = 79
3.1 Classical Definition of Consistency = 79
3.2 Definition of Strict (Nontrivial) Consistency = 82
3.2.1 Definition of Strict Consistency for the Pattern Recognition and the Regression Estimation Problems = 82
3.2.2 Definition or Strict Consistency for the Desity Estimation Problem = 84
3.3 Empirical Processes = 85
3.3.1 Remark on the Law of Large Numbers and Its Generalization = 86
3.4 The Key Theorem of Learning Theory (Theorem About Equivalence) = 88
3.5 Proof of the Key Theorem = 89
3.6 Strict Consistency of the Maximum Likelihood Method = 92
3.7 Necessary and Sufficient Conditions for Uniform Convergence of Frequencies to Their Probabilities = 93
3.7.1 Three Cases of Uniform Co 3.10 Kant's Problem of Demarcation and Popper's Theory of Nonfalsifability = 106nvergence = 93
3.7.2 Conditions of Uniform Convergence in the Simplest Model = 94
3.7.3 Entropy of a Set of Functions = 95
3.7.4 Theorem About Uniform Two-Sided Convergence = 97
3.8 Necessary and Sufficient Conditions for Uniform Convergence of Means to Their Expectations for a Set of Real-Valued Functions = 98
3.8.1 Entropy of a Set of Real-Valued Functions = 98
3.8.2 Theorem About Uniform Two-Sided Convergence = 99
3.9 Necessary and Sufficient Conditions for Uniform Convergence of Means to Their Expectations for Sets of Unbounded Functions = 100
3.9.1 Proof of Theorem 3.5 = 101
3.10 Kant's Problem of Demaractions and Popper's Theory of Nonfalsifiability = 106
3.11 Theorems About Nonfalsifiability = 108
3.11.1 Case of Complete Nonfalsifiability = 108
3.11.2 Theorem About Partial Nonfalsifiability = 109
3.11.3 Theorem About Potential Nonfalsifiability = 110
3.12 Conditions for One-Sided Uniform Convergence and Consistency of the Emprical Risk Minimization Principle = 112
3.13 Three Milestones in Learning Theory = 119
4 Bounds on the Risk for Indicator Loss Functions = 121
4.1 Bounds for the Simplest Model : Pessimistic Case = 122
4.1.1 The Simplest Model = 123
4.2 Bounds for the Simplest Modes : Optimistic Case = 125
4.3 Bounds for the Simplest Modes : General Case = 127
4.4 The Basic Inequalities : Pessimistic Case = 129
4.5 Proof of Theorem 4.1 = 131
4.5.1 The Basic Lemma = 131
4.5.2 Proof of Basic Lemma = 132
4.5.3 The Idea of Proving Theorem 4.1 = 134
4.5.4 Proof of Theorem 4.1 = 135
4.6 Basic Inequalities : General Case = 137
4.7 Proof of Theorem 4.2 = 139
4.8 Main Nonconstructive Bounds = 144
4.9 VC Dimension = 145
4.9.1 The Structure of the Growth Function = 145
4.9.2 Constructive Distribution-Free Bounds on Generalization Ability = 148
4.9.3 Solution of Generalized Glivenko-Cantelli Problem = 149
4.10 Proof of Theorem 4.3 = 150
4.11 Example of the VC Dimension of the Different Sets of Functions = 155
4.12 Remarks About the Bounds on the Generalization Ability of Learning Machines = 160
4.13 Bound on Deviation of Frequencies in Two Half-Samples = 163
Appendix to Chapter 4 : Lower Bounds on the Risk of the ERM Principle
A4.1 Two Strategies in Statistical Inference = 169
A4.2 Minimax Loss Strategy for Learning Problems = 171
A4.3 Upper Bounds on the Maximal Loss for the Empirical Risk Minimization Principle = 173
A4.3.1 Optimistic Case = 173
A4.3.2 Pessimistic Case = 174
A4.4 Lower Bound for the Minimax Loss Strategy in the Optimistic Case = 177
A4.5 Lower Bound for Minimax Loss Strategy for the Pessimistic Case = 179
5 Bounds on the Risk for Real-Valued Loss Functions = 183
5.1 Bounds for the Simplest Model : Pessimistic Case = 183
5.2 Concepts of Capacity for the Sets of Real-Valued Functions = 186
5.2.1 Nonconstructive Bounds on Generalization for Sets of Real-Valued Functions = 186
5.2.2. The Main Idea = 188
5.2.3 Concepts of Capacity for the Set of Real-Valued Functions = 190
5.3 Bounds for the General Model : Pessimistic Case = 192
5.4 The Basic Inequality = 194
5.4.1 Proof of Theorem 5.2 = 195
5.5 Bounds for the General Model : Universal Case = 196
5.5.1 Proof of Theorem 5.3 = 198
5.6 Bounds for Uniform Relative Convergence = 200
5.6.1 Proof of Theorem 5.4 for the Case p2 = 201
5.6.2 Proof of Theorem 5.4 for the Case 1 ≤ p = 204
5.7 Prior Information fot the Risk Minimization Problem in Sets of Unbounded Loss Functions = 207
5.8 Bounds on the Risk for Sets of Unbounded Nonnegative Functions = 210
5.9 Sample Selection and the Problem of Outliers = 214
5.10 The Main Results of the Theory of Bounds = 216
6 The Structural Risk Minimization Principle = 219
6.1 The Scheme of the Structural Risk Minimization Induction Principle = 219
6.1.1 Principle of Structural Risk Minimization = 221
6.2 Minimum Description Lengh and Structural Risk Minimization Inductive Principles = 224
6.2.1 The Idea About the Nature of Random Phenomena = 224
6.2.2 Minimum Description Lengh Principle for the Pattern Recognition Problem = 224
6.2.3 Bound for the Minimum Description Length Principle = 226
6.2.4 Structural Risk Minimization for the Simplest Model and Minimum Description Lengh Principle = 227
6.2.5 The Shortcoming of the Minimum Description Length Principle = 228
6.3 Consistency of the Structural Risk Minimization Principle and Asymptotic Bounds on the Rate of Convergence = 229
6.3.1 Proof of the Theorems = 232
6.3.2 Discussions and Example = 235
6.4 Bounds for the Regression Estimation Problem = 237
6.4.1 The Model of Regression Estimation Problem = 237
6.4.2 Proof of Theorem 6.4 = 241
6.5 The Problem of Approximating Functions = 246
6.5.1 Three Theorems of Classical Approximation Theory = 248
6.5.2 Curse of Dimensionality in Approximation Theory = 251
6.5.3 Problem of Approximation in Learning Theory = 252
6.5.4 The VC Dimension in Approximation Theory = 254
6.6 Problem of Local Risk Minimization = 257
6.6.1 Local Risk Minimization Model = 259
6.6.2 Bounds for the Local Risk Minimization Estimator = 262
6.6.3 Proofs of the Theorems = 265
6.6.4 Structural Risk Minimization Principle for Local Function EstimTION = 268
Appendix to Chapter 6 : Estimating Functions on the Basis of Indirect Measurements = 271
A6.1 Problems of Estimating the Results of Indirect Measurements = 271
A6.2 Theorems on Estimating Functions Using Indirect Measurements = 273
A6.3 Proofs of the Theorems = 276
A6.3.1 Proof of Theorem A6.1 = 276
A6.3.2 Proof of Theorem A6.2 = 281
A6.3.3 Proof of Theorem A6.3 = 283
7 Stochastic Ill － Posed Problems
7.1 Stochastic Ill － Posed Problems = 293
7.2 Regularization Method for Solving Stochastic Ill － Posed Problems = 297
7.3 Proofs of the Theorems = 299
7.3.1 Proof of Theorem 7.1 = 299
7.3.2 Proof of Theorem 7.2 = 302
7.3.3 Proof of Theorem 7.3 = 303
7.4 Conditions for Consistency of the Methods of Density Estimation = 305
7.5 Nonparametric Estimators of Density : Estimators Based on Approximations of the Distribution Function by an Empirical Distribution Funcion = 308
7.5.1 The Parzen Estimators = 308
7.5.2 Projection Estimators = 313
7.5.3 Spline Estimate of the Density. Approximation by Splines of the Odd Order = 313
7.5.4 Spline Estimate of the Density. Approximation by Splines of the Even Order = 314
7.6 Nonclassical Estimators = 315
7.6.1 Estimators for the Distribution Function = 315
7.6.2 Polygon Approximation of Distribution Function = 316
7.6.3 Kernel Density Estimator = 316
7.6.4 Projection Method of the Density Estimator = 318
7.7 Asymptotic Tate of Convergences fot Smooth Density Functions = 319
7.8 Proof of Theorem 7.4 = 322
7.9 Choosing a Value of Smoothing (Regularization) Parameter for the Problem of Density Estimation = 327
7.10 Estimation of the Ratio of Two Densities = 330
7.10.1 Estimation of Conditional Densities = 333
7.11 Estimation of Ratio of Two Densities on the Line = 334
7.12 Estimation of a Conditional Probability on a Line = 337
8 Estimating the Values of Function at Given Points = 339
8.1 The Scheme of Minimizing the Overall Risk = 339
8.2 The Method of Structural Minimization of the Overall Risk = 343
8.3 Bounds on the Uniform Relative Deviation of Frequencies in Tow Subsamples = 344
8.4 A Bound on the Uniform Relative Deviation of Means in Two Subsamples = 347
8.5 Estimation of Values of an Indicator Function in a Class of Linear Decision Rules = 350
8.6 Sample Selection for Estimating the Values of an Indicator Function = 355
8.7 Estimation of Values of a Real Function in the Class of Functions Linear in Their Parameters = 359
8.8 Sample Selection for Estimation of Values of Real-Valued Functions = 362
8.9 Local Algorithms for Estimating Values of an Indiator Function = 363
8.10 Local Algorithms for Estimating Values of a Real-Valued Function = 365
8.11 The Problem of Finding the Best Point in a Given Set = 367
8.11.1 Choice of the Most Probable Representative of the First Class = 368
8.11.2 Choice of the Best Point of a Given Set = 370
Ⅱ SUPPORT VECTOR ESTIMATION OF FUNCTOINS
9 Perceptrons and Their Generalizations = 375
9.1 Rosenblatt's Perceptron = 375
9.2 Proofs of the Theorems = 380
9.2.1 Proof of Novikoff Theorem = 380
9.2.2 Proof of Theorem 9.3 = 382
9.3 Method of Stochastic Approximation and Sigmoid Approximation of Indicator Functions = 383
9.3.1 Method of Stochastic Approximation = 384
9.3.2 Sigmoid Approximations of Indicator Functions = 385
9.4 Method of Potential Functions and Radial Basis Functions = 387
9.4.1 Method of Potential Functions in Asymptotic Learning Theory = 388
9.4.2 Radial Basic Function Method = 389
9.5 Three Theorems of Optimization Theory = 390
9.5.1 Fermat's Theorem (1629) = 390
9.5.2 Lagrange Multipliers Rule (1788) = 391
9.5.3 K u ·· hn-Tucker Theorem (1951) = 393
9.6 Neural Networks = 395
9.6.1 The Back-Propagation Method = 395
9.6.2 The Back-Propagation Algorithm = 398
9.6.3 Neural Networks for the Regression Estimation Problem = 399
9.6.4 Remarks on the Back-Propagation Method = 399
10 The Support Vector Method for Estimating Indicator Functions = 401
10.1 The Optimal Hyperplane = 401
10.2 The Optimal Hyperplane for Nonseparable Sets = 408
10.2.1 The Hard Margin Generalization of the Optimal Hyperplane = 408
10.2.2 The Basic Solution. Soft Margin Generalization = 411
10.3 Statistical Properties of the Optimal Hyperplane = 412
10.4 Proofs of the Theorems = 415
10.4.1 Proof of Theorem 10.3 = 415
10.4.2 Proof of Theorem 10.4 = 415
10.4.3 Leave-One-Out Procedure = 416
10.4.4 Proof of Theorem 10.5 and Theorem 9.2 = 417
10.4.5 Proof of Theorem 10.6 = 418
10.4.6 Proof of Theorem 10.7 = 421
10.5 The Idea of Support Vector Machine = 421
10.5.1 Generalization in High-Dimensional Space = 422
10.5.2 Hilbert-Schmidt Theory and Mercer Theorem = 423
10.5.3 Constructing SV Machines = 424
10.6 One More Approach to the Support Vector Method = 426
10.6.1 Minimizing the Number of Support Vectors = 426
10.6.2 Generalization for the Nonseparable Case = 427
10.6.3 Linear Optimization Method for SV Machines = 427
10.7 Selection of SV Machine Using Bounds = 428
10.8 Examples of SV Machines for Pattern Recognition = 430
10.8.1 Support Vector Method for Transductive Inference = 434
10.8.2 Radial Basis Function SV Machines = 431
10.8.3 Two-Layer Neural SV Machines = 432
10.9 Support Vector Method for Transductive Inference = 434
10.10 Multiclass Classification = 437
10.11 Remarks on Generalization of the SV Method = 440
11 The Support Vector Method for Estimating Real-Valued Functions = 443
11.1 ? -Insenstive Loss Functions = 443
11.2 Loss Functions for Robust Estimators = 445
11.3. Minimizing the Risk With ? -Insenstive Loss Functions = 448
11.3.1 Minimizing the Risk for a Fixed Element of the Structure = 449
11.3.2 The Basic Solutions = 452
11.3.3 Solution for the Huber Loss Function = 453
11.4 SV Machines for Function Estimation = 454
11.4.1 Minimiaing the Risk for a Fixed Element of the Structure in Feature Space = 455
11.4.2 The Basic Solutions in Feature Space = 456
11.4.3 Solution for Huber Loss Function Feature Space = 458
11.4.4 Linear Optimization Method = 459
11.4.5 Multi-Kernel Decomposition of Functions = 459
11.5 Constructing Kernels for Estimation of Real-Valud Functions = 460
11.5.1 Kernels Generating Expansion on Polynomials = 461
11.5.2 Constructing Multimensional Kernels = 462
11.6 Kernels Generating Splines = 464
11.6.1 Spline of Order a with a Finite Number of Knots = 464
11.6.2 Kernels Generating Splines with an Infinite Number of Knots = 465
11.6.3 Bd -Spine Approximations = 466
11.6.4 Bd Splines with an Infinite Number of Knots = 468
11.7 Kernels Generating Fourier Expansions = 468
11.7.1 Kernels for Regularized Fourier Expansions = 469
11.8 The Support Vector ANOVA Decomposition (SVAD) for Function Approximation and Regression Estimation = 471
11.9 SV Method for Solving Linear Operator Equations = 473
11.9.1 The SV Method = 473
11.9.2 Regularization by Choosing Parameters of ?i -Insensitivity = 478
11.10 SV Method of Density Estimation = 479
11.10.1 Spline Approximation of a Density = 480
11.10.2 Approximation of a Density with Gaussian Mixture = 481
11.11 Estimation of Conditional Probability and Conditional Density Function = 484
11.11.1 Estimation of Conditional Probability Functions = 484
11.11.2 Estimation of Conditional Density Functions = 488
11.12 Connection Between the SV Method and Sparse Function Approximation = 489
11.12.1 Reproducing Kernels Hilbert Spaces = 490
11.12.2 Modified Sparse Approximation an its Relation to SV Mahines = 491
12 SV Machines for Pattern Recognition
12.1 The Quadratic Optimization Problem = 493
12.1.1 Iterative Procedure for Specifying Support Vectors = 494
12.1.2 Methods for Solving the Reduced Optimization Problem = 496
12.2 Digit Recognition Problem. The U.S. Potal Service Database = 496
12.2.1 Performance for the U. S. Postal Service Database = 496
12.2.2 Some Important Details = 500
12.2.3 Comparison of Performance of the SV Machine with Gaussian Kernel to the Gaussian RBF Network = 503
12.2.4 The Best Results for U. S. Postal Service Database = 505
12.3 Tangent Distance = 506
12.4 Digit Recognition Problem. The NIST Database = 511
12.4.1 Performance for NIST Database = 511
12.4.2 Further Improvement = 512
12.4.3 The Best Results for NIST Database = 512
12.5 Future Racing = 512
12.5.1 One More Opportunity. The Transductive Inforence = 518
13 SV Machines for Function Approximations, Regression Estimation, and Signal Processing = 521
13.1 The Model Selection Problem = 521
13.1 1 Functional for Modedl Selection Based on the VC Bound = 522
13.1.2 Classical Functionals = 524
13.1.3 Experimental Omparison of Model Selection Methods = 525
13.1.4 The Problem of Feature Selectiion Has No General Solution = 526
13.2 Structure on the Set of Regularized Linear Function = 530
13.2.1 The L-Curve Method = 532
13.2.2 The Method of Effective Number of Parameters = 534
13.2.3 The Method of Effective VC Dimension = 536
13.2.4 Experiments on Measuring the Effectie VC Dimension = 540
13.3 Function Approximation Using the SV Method = 543
13.3.1 Why Does the Value of $$\varepsilon$$ Control the Number of support Vectors? = 546
13.4 SV Machine for Regression Estimation = 549
13.4.1 Problem of Data Smoothing = 549
13.4.2 Estimation of Linear Regression Functions = 550
13.4.3 Estimation of Nonlinear Regression Function = 556
13.5 SV Method for Solving the Positron Emission Tomography (PET) Problem = 558
13.5.1 Description of PET = 558
13.5.2 Problem of Solving the Radon Equation = 560
13.5.3 Generalization of the Residual Principle of Solvint PET Problems = 561
13.5.4 The Classical Methods of Solving the PET Problem = 562
13.5.5 The SV Method for Solving the PET Problem = 563
13.6 Remark About the SV Method = 567
Ⅲ STATISTICAL FOUNDATION OF LEARNING THEORY
14 Necessary and Sufficient Conditions for Uniform Convergence of Frequencies to Their Probilites = 571
14.1 Uniform Convergency of Frequencies to their Probalities = 572
14.2 Basic Lemma = 573
14.3 Entropy of the Set of Events = 576
14.4 Asymptotic : Properties of the Entropy = 578
14.5 Necessary and Sufficient Conditions of Uniform Convergence Proof of Suffiencty = 584
14.6 Necessary and Sufficient Conditions. Continuation of Proving Necessity = 592
15 Necessary and Sufficient Conditions for Uniform Convergence of Means to Their Expections = 597
15.1 ? Entropy = 597
15.1.1 Proof of the Existence of the Limit = 600
15.1.2 Proof of the Convergence of the Sequence = 601
15.2 The Quasicube = 603
15.3 /varepsilon of a Set = 608
15.4 An Auxiliary Lemma = 610
15.5 Necessary and Sufficient Conditions for Uniform Convergence. The Proof of Necessity = 614
15.6 Necessary and Sufficient Conditions for Uniform Convergence. The Proof of Suifficiency = 618
15.7 Corollaries from Theorem 15.1 = 624
16 Necessary and Sufficient Conditions for Uniform One-Sided Convergence or Means to Their Expectations = 629
16.1 Introduction = 629
16.2 Maximum Volume Sections = 630
16.3 The Theorem on the Average Logarith = 636
16.4 Theorem on the Existence of a Corridor = 642
16.5 Theorem on the Existence of Functions Closer to the Corridor Boundaries (Theorem on Potential Nonfalsifability) = 650
16.6 The Necessary Conditions = 660
16.7 The Necessary and Sufficient Conditions = 666
Comments and Bibliographical Remars = 681
References = 723
Index = 733

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