Anti-symmetric Barron functions and their approximation with sums of determinants

TL;DR

This paper demonstrates that anti-symmetric functions in the Barron space can be efficiently approximated using sums of determinants, offering a factorial complexity improvement and explaining the success of determinant-based neural architectures in quantum chemistry.

Contribution

The paper introduces a novel approximation method for anti-symmetric Barron functions using sums of determinants, providing theoretical complexity improvements.

Findings

Efficient approximation of anti-symmetric functions with sums of determinants.

Factorial reduction in complexity over standard Barron space representations.

Theoretical justification for determinant-based architectures in quantum chemistry.

Abstract

A fundamental problem in quantum physics is to encode functions that are completely anti-symmetric under permutations of identical particles. The Barron space consists of high-dimensional functions that can be parameterized by infinite neural networks with one hidden layer. By explicitly encoding the anti-symmetric structure, we prove that the anti-symmetric functions which belong to the Barron space can be efficiently approximated with sums of determinants. This yields a factorial improvement in complexity compared to the standard representation in the Barron space and provides a theoretical explanation for the effectiveness of determinant-based architectures in ab-initio quantum chemistry.