Compressive sensing and truncated moment problems on spheres

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 ${\mu }^{\ast }$ supported on $C$ and we prove that ${\mu }^{\ast }$ 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 $\mu ={\mu }^{\ast }$ when $\mu$ is supported on $C$ and prove that ${\mu }^{\ast }$ is close to the best approximation to $\mu$ supported on $C$ provided that all points in the support of $\mu$ are close to $C$.