arXiv:2503.04263

Bi-Lipschitz Ansatz for Anti-Symmetric Functions

Nadav Dym, Jianfeng Lu, Matan Mizrachi

Published Mar 6, 2025• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2503.04263

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arXiv:2503.04263

Bi-Lipschitz Ansatz for Anti-Symmetric Functions

Nadav Dym, Jianfeng Lu, Matan Mizrachi

Published Mar 6, 2025• 0 citations

arXiv HTML PDF

Abstract

ε

. We also provide preliminary experimental evidence suggesting improved performance in learning antisymmetric functions.

DOI: 10.48550/arxiv.2503.04263

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ε

. We also provide preliminary experimental evidence suggesting improved performance in learning antisymmetric functions.

ε

. While the approximation still suffers from the curse of dimensionality, to the best of our knowledge, these are the first results in the literature with explicit error bounds for functions with symmetry or anti-symmetry constraints.

We study separations between two fundamental models (or \emph{Ansätze}) of antisymmetric functions, that is, functions

f

of the form

f(x_{σ(1)}, \ldots, x_{σ(N)}) = \text{sign}(σ)f(x_1, \ldots, x_N)

, where

σ

is any permutation. These arise in the context of quantum chemistry, and are the basic modeling tool for wavefunctions of Fermionic systems. Specifically, we consider two popular antisymmetric Ansätze: the Slater representation, which leverages the alternating structure of determinants, and the Jastrow ansatz, which augments Slater determinants with a product by an arbitrary symmetric function. We construct an antisymmetric function in

N

dimensions that can be efficiently expressed in Jastrow form, yet provably cannot be approximated by Slater determinants unless there are exponentially (in

N^2

) many terms. This represents the first explicit quantitative separation between these two Ansätze.

\widetilde{O}(N^2)

Representation of General Continuous Anti-symmetric Function

{\widetilde O}(N^2)

ansatz based on permutation-equivariant functions. We prove that our ansatz can represent any AS continuous functions, and can accommodate the determinant-based structure proposed by Hutter [14], solving the proposed open problems that

{O}(N)

Slater determinants are sufficient to provide universal representation of AS continuous functions. Together, we offer a generalizable and efficient approach to representing AS continuous functions, shedding light on designing neural networks to learn wave functions.

O(N^{3N-3})

determinants to represent a generic totally antisymmetric polynomial. Our proof is based on bounding the dimension of the source of the ansatz from above and bounding the dimension of the target from below.