Bi-Lipschitz Ansatz for Anti-Symmetric Functions

TL;DR

The paper tackles robustly representing antisymmetric functions, essential for quantum many-body wavefunctions, by constructing a bi-Lipschitz, $A_n$-invariant map $F$ that enables stable antisymmetric learning. It first provides a concrete $d=1$ construction using a sorting-based isometry and an antisymmetric component, then generalizes to higher dimensions by stacking multiple copies of a bi-Lipschitz map, guaranteeing injectivity and bi-Lipschitzness up to $A_n$ with $m=2nd+1$ channels. The authors prove quantitative approximation guarantees: any 1-Lipschitz antisymmetric function on a compact $A_n$-invariant set can be approximated to within $\varepsilon$ by an MLP with $O(\varepsilon^{-m/2})$ parameters, and discuss how intrinsic dimension could improve the rate further. Empirical results on the determinant task show the proposed bi-Lipschitz ansatz outperforms Vandermonde-based and naive MLP baselines in accuracy and convergence, suggesting practical benefits for learning antisymmetric quantum states.

Abstract

Motivated by applications for simulating quantum many body functions, we propose a new universal ansatz for approximating anti-symmetric functions. The main advantage of this ansatz over previous alternatives is that it is bi-Lipschitz with respect to a naturally defined metric. As a result, we are able to obtain quantitative approximation results for approximation of Lipschitz continuous antisymmetric functions. Moreover, we provide preliminary experimental evidence to the improved performance of this ansatz for learning antisymmetric functions.

Figures (1)

• Figure 1: Training-Validation convergence comparison of our Ansatz (black), against ye2024widetilde's (red) and an MLP (green), for approximating the determinant of matrices of different orders (denoted by $n$). Our ansatz exhibits faster convergence and better or comparable accuracy.

Theorems & Definitions (6)

• Proposition 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof