arXiv subjects
Publications and source records attributed to Hernán García.
We study the problem of reconstructing a positive discrete measure on a compact set $K \subseteq \mathbb{R}^n$ from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.
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 $C$.