Compressive sensing and truncated moment problems on spheres

Abstract

We propose convex optimization algorithms to recover a good approximation of a point measure $μ$ 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 $μ^*$ supported on $C$ and we prove that $μ^*$ is a good approximation to $μ$ 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 $μ=μ^*$ when $μ$ is supported on $C$ and prove that $μ^*$ is close to the best approximation to $μ$ supported on $C$ provided that all points in the support of $μ$ are close to $C$.