Machine learning and invariant theory

TL;DR

This paper discusses methods for explicitly parameterizing equivariant functions in machine learning using invariant theory, focusing on polynomial and smooth maps under group actions, with applications inspired by physical constraints.

Contribution

It introduces a general procedure based on Malgrange's method for parameterizing equivariant functions, applicable to polynomial and smooth maps under group actions.

Findings

Provides explicit parameterization methods for equivariant polynomial maps.

Extends parameterization techniques to smooth maps for compact Lie groups.

Connects invariant theory with practical machine learning applications.

Abstract

Inspired by constraints from physical law, equivariant machine learning restricts the learning to a hypothesis class where all the functions are equivariant with respect to some group action. Irreducible representations or invariant theory are typically used to parameterize the space of such functions. In this article, we introduce the topic and explain a couple of methods to explicitly parameterize equivariant functions that are being used in machine learning applications. In particular, we explicate a general procedure, attributed to Malgrange, to express all polynomial maps between linear spaces that are equivariant under the action of a group $G$, given a characterization of the invariant polynomials on a bigger space. The method also parametrizes smooth equivariant maps in the case that $G$ is a compact Lie group.