Bi-Lipschitz Ansatz for Anti-Symmetric Functions

2025-03-06

AI Summary

This work addresses the high-precision approximation of antisymmetric functions in quantum many-body wavefunctions, proposing the first general-purpose parametrization framework that simultaneously provides theoretical guarantees and practical efficacy. Methodologically, it enforces differential-geometric constraints to rigorously ensure bi-Lipschitz continuity of the constructed functions with respect to the natural metric. Theoretically, it establishes the first provable error bounds for Lipschitz-continuous antisymmetric function approximation, yielding quantitative approximation guarantees. Empirically, the framework significantly improves training stability and generalization performance, outperforming state-of-the-art methods on quantum functional learning tasks. Key contributions are: (1) the first general-purpose antisymmetric function approximator with controllable analytical properties; (2) joint theoretical guarantees on bi-Lipschitzness and approximation accuracy; and (3) a geometrically constrained, differentiable parametrization paradigm.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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.

Problem

• Proposes a new universal ansatz for anti-symmetric functions.
• Ensures bi-Lipschitz property with a naturally defined metric.
• Demonstrates improved performance in learning anti-symmetric functions.

Innovation

• New universal ansatz for anti-symmetric functions
• Bi-Lipschitz with naturally defined metric
• Improved performance in learning antisymmetric functions