Compressive sensing and truncated moment problems on spheres

TL;DR

This paper introduces convex optimization algorithms for approximating measures on spheres from moments, demonstrating their effectiveness with random polynomial samples and providing conditions for exact recovery and approximation.

Contribution

It develops convex optimization methods for measure recovery on spheres using random polynomial moments, with theoretical guarantees for exact and approximate solutions.

Findings

Algorithms successfully recover measures supported on finite sets.

Conditions established for exact measure recovery.

Approximate solutions are close to true measures under certain proximity conditions.

Abstract

We propose convex optimization algorithms to recover a good approximation of a point measure $\mu$ on the unit sphere $S\subseteq \mathbb{R}^n$ from its moments with respect to a set of real-valued functions $f_1,\dots, f_m$. Given a finite subset $C\subseteq S$ the algorithm produces a measure $\mu^*$ supported on $C$ and we prove that $\mu^*$ is a good approximation to $\mu$ whenever the functions $f_1,\dots, f_m$ are a sufficiently large random sample of independent Kostlan-Shub-Smale polynomials. More specifically, we give sufficient conditions for the validity of the equality $\mu=\mu^*$ when $\mu$ is supported on $C$ and prove that $\mu^*$ is close to the best approximation to $\mu$ supported on $C$ provided that all points in the support of $\mu$ are close to $C$.