Algebraic Geometry and Statistical Learning Theory

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Hardcover | 13 August 2009

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298 Pages

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22.86 x 15.24 x 2.06

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Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established: 1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models; 2. the asymptotic behaviour of the marginal likelihood or 'the evidence' is derived based on zeta function theory; 3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors; 4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties.

Industry Reviews

"Overall, the many insightful remarks and simple direct language make the book a pleasure to read." Shaowei Lin, Mathematical Reviews

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You Can Find This Book In

Non-Fiction Computing & I.T.Computer Science Artificial Intelligence Computer Vision Mathematics Geometry Algebraic Geometry Machine Learning Pattern Recognition

Preface	p. vii
Introduction	p. 1
Basic concepts in statistical learning	p. 1
Statistical models and learning machines	p. 10
Statistical estimation methods	p. 18
Four main formulas	p. 26
Overview of this book	p. 41
Probability theory	p. 42
Singularity theory	p. 48
Polynomials and analytic functions	p. 48
Algebraic set and analytic set	p. 50
Singularity	p. 53
Resolution of singularities	p. 58
Normal crossing singularities	p. 66
Manifold	p. 72
Algebraic geometry	p. 77
Ring and ideal	p. 77
Real algebraic set	p. 80
Singularities and dimension	p. 86
Real projective space	p. 87
Blow-up	p. 91
Examples	p. 99
Zeta function and singular integral	p. 105
Schwartz distribution	p. 105
State density function	p. 111
Mellin transform	p. 116
Evaluation of singular integral	p. 118
Asymptotic expansion and b-function	p. 128
Empirical processes	p. 133
Convergence in law	p. 133
Function-valued analytic functions	p. 140
Empirical process	p. 144
Fluctuation of Gaussian processes	p. 154
Singular learning theory	p. 158
Standard form of likelihood ratio function	p. 160
Evidence and stochastic complexity	p. 168
Bayes and Gibbs estimation	p. 177
Maximum likelihood and a posteriori	p. 203
Singular learning machines	p. 217
Learning coefficient	p. 217
Three-layered neural networks	p. 227
Mixture models	p. 230
Bayesian network	p. 233
Hidden Markov model	p. 234
Singular learning process	p. 235
Bias and variance	p. 239
Non-analytic learning machines	p. 245
Singular statistics	p. 249
Universally optimal learning	p. 249
Generalized Bayes information criterion	p. 252
Widely applicable information criteria	p. 253
Singular hypothesis test	p. 258
Realization of a posteriori distribution	p. 264
From regular to singular	p. 274
Bibliography	p. 277
Index	p. 284
Table of Contents provided by Ingram. All Rights Reserved.

Algebraic Geometry and Statistical Learning Theory

At a Glance

Hardcover

Industry Reviews

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How to return your order

Defective items

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