  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.
