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Algebraic Machine Learning with an Application to Chemistry

Ezzeddine El Sai, Parker Gara, Markus J. Pflaum

TL;DR

A machine learning pipeline that captures fine-grain geometric information without having to rely on any smoothness assumptions is developed and a heuristic for numerically detecting points lying near the singular locus of the underlying variety is proposed.

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 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.

Algebraic Machine Learning with an Application to Chemistry

TL;DR

A machine learning pipeline that captures fine-grain geometric information without having to rely on any smoothness assumptions is developed and a heuristic for numerically detecting points lying near the singular locus of the underlying variety is proposed.

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 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.
Paper Structure (11 sections, 9 theorems, 40 equations, 10 figures, 5 algorithms)

This paper contains 11 sections, 9 theorems, 40 equations, 10 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background in Real Algebraic Geometry
  3. Ordered Fields
  4. The Nullstellensatz
  5. Properties of Algebraic Varieties
  6. Learning the Underlying Variety
  7. Expanding on the MAP Model
  8. Algebraic Computations
  9. Numerical Computations
  10. Results
  11. Related Work

Key Result

Proposition 2.2

Let $\mathbb{K}$ be a subfield of $\mathbb{R}$. If $S$ is a non-empty subset of the polynomial ring $\mathbb{K}[x_{1},\ldots ,x_{n}]$, then $Z(S)$ can be written as $Z(f)$ for a single polynomial $f\in \mathbb{K}[x_{1},\ldots ,x_{n}]$.

Figures (10)

  • Figure 1.1:
  • Figure 2.1: The union of a sphere and a plane as a variety
  • Figure 6.1: Samples from $V$ and from the learned variety
  • Figure 6.2: Samples from the singular locus of $V$ and results from the singularity heuristic
  • Figure 6.3: MAP model and singularity heuristic performance (noise-free)
  • ...and 5 more figures

Theorems & Definitions (17)

  • Proposition 2.2
  • proof
  • Theorem 2.4: Hilbert's Nullstellensatz
  • Theorem 2.5: Real Nullstellensatz Neuhaus1998
  • Remark 2.6
  • Theorem 2.7
  • Proposition 2.9
  • Proposition 2.10
  • proof
  • Theorem 2.11
  • ...and 7 more