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

#### Table of Contents

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