Algebraic Machine Learning with an Application to Chemistry
This addresses the challenge of handling data with singularities in fields like chemistry, offering a novel approach for scientific applications, though it appears incremental as it builds on existing geometric and algebraic concepts.
The paper tackles the problem of analyzing complex scientific datasets where traditional topological and geometric methods are limited by coarse information or smoothness assumptions, by developing a machine learning pipeline based on algebraic geometry that captures fine-grain geometric details without relying on smoothness, resulting in a method that finds underlying varieties through MAP optimization and eigenvalue computations.
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 Gröbner 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.