Algebraic Machine Learning with an Application to Chemistry

TL;DR

This paper introduces an algebraic geometry-based machine learning approach that captures detailed geometric features of data, especially in complex physical models with singularities, without relying on smoothness assumptions.

Contribution

It develops a novel pipeline using algebraic varieties and eigenvalue optimization to learn and analyze the geometric structure of data, including singularities.

Findings

Successfully models complex data geometries without smoothness assumptions

Uses eigenvalue computations to identify underlying algebraic varieties

Proposes heuristic methods for detecting singular points

Abstract

As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in topological tools such as persistent homology. However, on the one hand, topological tools are inherently limited to providing only coarse information about the underlying space of the data. On the other hand, more geometric approaches rely predominately on the manifold hypothesis, which asserts that the underlying space is a smooth manifold. This assumption fails for many physical models where the underlying space contains singularities. In this paper we develop a machine learning pipeline that captures fine-grain geometric information without having to rely on any smoothness assumptions. Our approach involves working within the scope of algebraic geometry and algebraic varieties instead of differential geometry and smooth manifolds. In the setting of the variety hypothesis, the learning problem becomes to find the underlying variety using sample data. We cast this learning problem into a Maximum A Posteriori optimization problem which we solve in terms of an eigenvalue computation. Having found the underlying variety, we explore the use of Gr\"obner bases and numerical methods to reveal information about its geometry. In particular, we propose a heuristic for numerically detecting points lying near the singular locus of the underlying variety.