Machine learning and invariant theory
This work addresses the problem of incorporating symmetry constraints from physical laws into machine learning models, providing foundational tools for researchers in equivariant ML, but it is incremental as it explicates existing methods.
The paper introduces methods for parameterizing equivariant functions in machine learning, focusing on a general procedure to express all polynomial maps between linear spaces that are equivariant under group actions, with extensions to smooth maps for compact Lie groups.
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.