Algebraic Geometry and Statistical Learning Theory

Sumio Watanabe (Tokyo Institute of Technology)

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English

Cambridge University Press
13 August 2009

Mathematics & Sciences; Algebraic geometry; Machine learning

Series: Cambridge Monographs on Applied and Computational Mathematics

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.

By: Sumio Watanabe (Tokyo Institute of Technology)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 233mm, Width: 155mm, Spine: 20mm
Weight: 560g
ISBN: 9780521864671
ISBN 10: 0521864674
Series: Cambridge Monographs on Applied and Computational Mathematics
Publication Date: 13 August 2009
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active