Polynomial-time Sparse Measure Recovery: From Mean Field Theory to Algorithm Design

TL;DR

This paper uses mean field theory to inspire a new polynomial-time algorithm for sparse measure recovery from Fourier moments, outperforming convex relaxations in certain regimes and applying to neural network optimization.

Contribution

It introduces a novel algorithm for sparse measure recovery based on mean field theory, bridging theoretical physics insights with practical algorithm design.

Findings

Improves recovery over convex relaxation methods in specific regimes

Applies the algorithm to neural network optimization tasks

Demonstrates polynomial-time recovery for sparse measures over real numbers

Abstract

Mean field theory has provided theoretical insights into various algorithms by letting the problem size tend to infinity. We argue that the applications of mean-field theory go beyond theoretical insights as it can inspire the design of practical algorithms. Leveraging mean-field analyses in physics, we propose a novel algorithm for sparse measure recovery. For sparse measures over $\mathbb{R}$, we propose a polynomial-time recovery method from Fourier moments that improves upon convex relaxation methods in a specific parameter regime; then, we demonstrate the application of our results for the optimization of particular two-dimensional, single-layer neural networks in realizable settings.