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

arXiv: https://arxiv.org/abs/2303.12856 · HTML: https://mlsanity.com/paper/2303.12856/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 22 Mar 2023.

## Builds on (most cited first, 10 of 10)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [A simple neural network module for relational reasoning](https://mlsanity.com/paper/1706.01427/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [A Functional Perspective on Learning Symmetric Functions with Neural Networks](https://mlsanity.com/paper/2008.06952/) (2020)
- [Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states](https://mlsanity.com/paper/2008.00118/) (2020)
- [Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation](https://mlsanity.com/paper/2112.03491/) (2021)
- [On Representing (Anti)Symmetric Functions](https://mlsanity.com/paper/2007.15298/) (2020)
- [Towards Antisymmetric Neural Ansatz Separation](https://mlsanity.com/paper/2208.03264/) (2022)

## Built on by (most cited first, 2 of 2)

- [Statistical Learning Guarantees for Group-Invariant Barron Functions](https://mlsanity.com/paper/2509.23474/) (2025)
- [Uniform Ck Approximation of G-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations](https://mlsanity.com/paper/2403.01339/) (2024)

## Abstract (arXiv)

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.