An Approach with Toric Varieties for Singular Learning Machines

TL;DR

This paper explores the use of toric varieties and Hilbert bases from algebraic geometry to resolve singularities in statistical learning machines, particularly within the framework of Singular Learning Theory.

Contribution

It introduces a novel application of Hilbert bases for resolving singularities in learning machines using toric varieties, linking algebraic geometry with statistical learning.

Findings

Formalizes singular learning machines as toric varieties

Proves a theorem characterizing singular machines via toric resolution

Reproduces key results in Singular Statistical Learning

Abstract

The Computational Algebraic Geometry applied in Algebraic Statistics; are beginning to exploring new branches and applications; in artificial intelligence and others areas. Currently, the development of the mathematics is very extensive and it is difficult to see the immediate application of few theorems in different areas, such as is the case of the Theorem 3.9 given in [10] and proved in part of here. Also this work has the intention to show the Hilbert basis as a powerful tool in data science; and for that reason we compile important results proved in works by, S. Watanabe [27], D. Cox, J. Little and H. Schenck [8], B. Sturmfels [16] and G. Ewald [10]. In this work we study, first, the fundamental concepts in Toric Algebraic Geometry. The principal contribution of this work is the application of Hilbert basis (as one realization of Theorem 3.9) for the resolution of singularities with toric varieties, and a background in Lattice Polytope. In the second part we apply this theorem to problems in statistical learning, principally in a recent area as is the Singular Learning Theory. We define the singular machines and the problem of Singular Learning through the computing of learning curves on these statistical machines. We review and compile results on the work of S. Watanabe in Singular Learning Theory, ref.; [17], [20], [21], also revising the important result in [26], about almost the machines are singular, we formalize this theory withtoric resolution morphism in a theorem proved here (Theorem 5.4), characterizing these Learning Machines as toric varieties, and we reproduce results previously published in Singular Statistical Learning seen in [19], [20], [23].