# Bi-Lipschitz Ansatz for Anti-Symmetric Functions

arXiv: https://arxiv.org/abs/2503.04263 · HTML: https://mlsanity.com/paper/2503.04263/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 32 references that are in the map.
- First posted on arXiv 6 Mar 2025.

## Builds on (most cited first, 12 of 12)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem](https://mlsanity.com/paper/2306.06529/) (2023)
- [Permutation Invariant Representations with Applications to Graph Deep Learning](https://mlsanity.com/paper/2203.07546/) (2022)
- [On Representing (Anti)Symmetric Functions](https://mlsanity.com/paper/2007.15298/) (2020)

## Built on by (most cited first, 3 of 3)

- [Quantitative Bounds for Sorting-Based Permutation-Invariant Embeddings](https://mlsanity.com/paper/2510.22186/) (2025)
- [Toward bilipshiz geometric models](https://mlsanity.com/paper/2511.11735/) (2025)
- [Quantitative Approximation Rates for Group Equivariant Learning](https://mlsanity.com/paper/2602.20370/) (2026)

## Abstract (arXiv)

Motivated by applications to the simulation of quantum many-body systems by neural networks, researchers have suggested several models which are antisymmetric by construction, and can approximate all antisymmetric functions. However, these works either require very high computational complexity to attain universal approximation, or suffer from discontinuities. In this paper, we introduce two antisymmetric ansatzes which do not suffer from these disadvantages. The first is based on a bi-Lipschitz embedding with respect to a naturally defined metric. The second is a modular anti-symmetrizing projection framework based on the frame-averaging methodology. Both approaches yield continuous antisymmetric models which attain universal approximation guarantees with a polynomial complexity in problem size. Moreover, for both approaches, we obtain quantitative approximation results that bound the number of parameters required to approximate Lipschitz antisymmetric functions to a given accuracy ε. We also provide preliminary experimental evidence suggesting improved performance in learning antisymmetric functions.