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.

翻译：平均场理论通过让问题规模趋于无穷，为各类算法提供了理论洞见。我们认为平均场理论的应用不仅限于理论洞见，更能激发实用算法的设计。借助物理学中的平均场分析，我们提出了一种用于稀疏测度恢复的新算法。针对定义在$\mathbb{R}$上的稀疏测度，我们提出了一种基于傅里叶矩的多项式时间恢复方法，该方法在特定参数区间内优于凸松弛方法；随后，我们展示了该结果在可学习设定下对特定二维单层神经网络优化的应用。