By Sumio Watanabe

ISBN-10: 0521864674

ISBN-13: 9780521864671

Bound to be influential, Watanabe's e-book lays the rules for using algebraic geometry in statistical studying conception. Many models/machines are singular: combination types, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the idea accomplished right here underpins exact estimation innovations within the presence of singularities.

**Read or Download Algebraic Geometry and Statistical Learning Theory PDF**

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**Additional info for Algebraic Geometry and Statistical Learning Theory**

**Sample text**

In Chapter 5, we study the convergence in law of the empirical process to a Gaussian process, ξn (u) → ξ (u). This is the central limit theorem on the functional space. Also we introduce the partial integral on the function space. Based on mathematical foundations in chapters 2, 3, 4, and 5, the four main formulas are rigorously proved in Chapter 6. These are generalizations of the conventional statistical theory of regular models to singular models. We find two birational invariants, the maximum pole of the zeta function and the singular fluctuation, which determine the statistical learning process.

The origin is a critical point of the function f (x, y, z) = (xy + z)2 . (3) In the set A = {(x, y); xy = 0}, the origin is a singularity of A, which is a critical point of the function f (x, y) = xy. The singular locus is Sing(A) = {(0, 0)}. (4) In the set A = {(x, y); y 2 − x 3 = 0}, the origin is a singularity of A, which is a critical point of the function f (x, y) = y 3 − x 2 . (5) In the set A = {(x, y); x 5 − y 3 = 0}, the origin is a singularity of A, which is a critical point of f (x, y) = x 5 − y 3 .

Definition of a nonsingular point there is no local maximum or minimum point. The origin (0, 0) is a critical point of f . If |x| ≥ |y|, then f (x, y) ≥ 0, and if |x| ≤ |y|, then f (x, y) ≤ 0. Such a critical point is said to be a saddle point. 5 (C r Isomorphism) Let U , V be open sets of the real Euclidean space Rd . If there exists a one-to-one map f : U → V such that both f and f −1 are functions of C r class, then U is said to be C r isomorphic to V , and f is called a C r isomorphism. If both f and f −1 are analytic functions, then U is said to be analytically isomorphic to V and f is called an analytic isomorphism.