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Exact and Efficient Representation of Totally Anti-Symmetric Functions

Ziang Chen, Jianfeng Lu

TL;DR

The paper tackles the challenge of representing totally anti-symmetric functions in high dimensions by introducing an ansatz that composes a continuous odd function with a fixed set of anti-symmetric basis functions η. It proves a representation theorem: any anti-symmetric and continuous f on Ω^n can be written uniquely as f(x) = g(η(x)) with g continuous and odd, reducing the problem to learning an odd function on η(Ω^n). An explicit, polynomial-scaling construction of η is provided, along with a characterization of the singular locus Sing_{d,n} where two inputs collide, and the limitations of regularity in the resulting g, demonstrated via Lipschitz and C^1 counterexamples. These results offer a robust, dimensionally scalable framework for anti-symmetric function representation, with implications for efficient neural representations in quantum many-body contexts and beyond.

Abstract

This paper concerns the long-standing question of representing (totally) anti-symmetric functions in high dimensions. We propose a new ansatz based on the composition of an odd function with a fixed set of anti-symmetric basis functions. We prove that this ansatz can exactly represent every anti-symmetric and continuous function and the number of basis functions has efficient scaling with respect to dimension (number of particles). The singular locus of the anti-symmetric basis functions is precisely identified.

Exact and Efficient Representation of Totally Anti-Symmetric Functions

TL;DR

The paper tackles the challenge of representing totally anti-symmetric functions in high dimensions by introducing an ansatz that composes a continuous odd function with a fixed set of anti-symmetric basis functions η. It proves a representation theorem: any anti-symmetric and continuous f on Ω^n can be written uniquely as f(x) = g(η(x)) with g continuous and odd, reducing the problem to learning an odd function on η(Ω^n). An explicit, polynomial-scaling construction of η is provided, along with a characterization of the singular locus Sing_{d,n} where two inputs collide, and the limitations of regularity in the resulting g, demonstrated via Lipschitz and C^1 counterexamples. These results offer a robust, dimensionally scalable framework for anti-symmetric function representation, with implications for efficient neural representations in quantum many-body contexts and beyond.

Abstract

This paper concerns the long-standing question of representing (totally) anti-symmetric functions in high dimensions. We propose a new ansatz based on the composition of an odd function with a fixed set of anti-symmetric basis functions. We prove that this ansatz can exactly represent every anti-symmetric and continuous function and the number of basis functions has efficient scaling with respect to dimension (number of particles). The singular locus of the anti-symmetric basis functions is precisely identified.
Paper Structure (7 sections, 6 theorems, 24 equations)

This paper contains 7 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Representation theorem for anti-symmetric functions
  4. Validation of Assumption \ref{['asp:eta']}
  5. Singular locus of the anti-symmetric basis functions
  6. Counterexample for Lipschitz continuity
  7. Counterexample for continuous differentiability

Key Result

Theorem 1

Given $d,n\geq 1$, let $\Omega\subset\mathbb{R}^d$ be compact and let $\bm{\eta}:\Omega^n\to\mathbb{R}^m$ satisfy Assumption asp:eta. For any anti-symmetric continuous function $f:\Omega^n\to\mathbb{R}$, there exists a unique $g:\bm{\eta}(\Omega^n)\to \mathbb{R}$ that is continuous and odd, and sati where $\bm{\eta}(\Omega^n)$ is equipped with the topology induced from $\mathbb{R}^m$.

Theorems & Definitions (15)

  • Remark 2.1
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • proof : Proof of Proposition \ref{['prop:construct_eta']}
  • ...and 5 more