## Algebraic Geometry and Statistical Learning Theory by Sumio Watanabe

By Sumio Watanabe

Certain to be influential, Watanabe's publication lays the rules for using algebraic geometry in statistical studying idea. Many models/machines are singular: blend types, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the idea completed right here underpins actual estimation thoughts within the presence of singularities.

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Readers who are familiar with probability theory can skip this section. 11 (Metric space) Let D: × be a set. A function D (x, y) → D(x, y) ∈ R is called a metric if it satisfies the following three conditions. (1) For arbitrary x, y ∈ , D(x, y) = D(y, x) ≥ 0. (2) D(x, y) = 0 if and only if x = y. (3) For arbitrary x, y, z ∈ , D(x, y) + D(y, z) ≥ D(x, z). A set with a metric is called a metric space. The set of open neighborhoods of a point x ∈ is defined by {U (x); > 0} where U (x) = {y ∈ ; D(x, y) < }.

Hence if X and Y have the same probability distribution, we can predict E[Y ] based on the information of E[X]. (3) In statistical learning theory, it is important to predict the expectation value of the generalization error from the training error. (4) If E[|X|] = C then, for arbitrary M > 0, C = E[|X|] ≥ E[|X|]{|X|>M} ≥ ME[1]{|X|>M} = MP (|X| > M). Hence C , M which is well known as Chebyshev’s inequality. The same derivation is often effective in probability theory. P (|X| > M) ≤ 46 Introduction (5) The following conditions are equivalent.

4 (Critical point of a function) Let U be an open set of Rd , and f : U → R1 be a function of C 1 class. (1) A point x ∗ ∈ U is called a critical point of f if it satisfies ∇f (x ∗ ) = 0. If x ∗ is a critical point of f , then f (x ∗ ) is called a critical value. (2) If there exists an open set U ⊂ U such that x ∗ ∈ U and f (x) ≤ f (x ∗ ) (∀x ∈ U ), then x ∗ is called a local maximum point of f . If x ∗ is a local maximum point, then f (x ∗ ) is called a local maximum value. (3) If there exists an open set U ⊂ U such that x ∗ ∈ U and f (x) ≥ f (x ∗ ) (∀x ∈ U ), then x ∗ is called a local minimum point of f .