Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry

TL;DR

This paper introduces symmetric and antisymmetric kernels tailored for quantum physics and chemistry, analyzing their properties, feature space dimensions, and proposing efficient representations to leverage symmetries for improved learning.

Contribution

It develops new symmetric and antisymmetric kernels, proves their density in relevant function spaces, and introduces a Slater determinant representation for efficient computation.

Findings

Symmetric and antisymmetric kernels have dense RKHS in their respective function spaces.

Exploiting symmetries reduces training data requirements.

Proposed kernels are effective in quantum physics and chemistry applications.

Abstract

We derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exploiting symmetries or antisymmetries the size of the training data set can be significantly reduced. The results are illustrated with guiding examples and simple quantum physics and chemistry applications.