Compressive sensing and truncated moment problems on spheres

2017/10/25 by García, Hernán, Hernández, Camilo, Junca, Mauricio +1
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1710.09496

Abstract

We propose convex optimization algorithms to recover a good approximation of a point measure μ on the unit sphere S⊆ ℝⁿ from its moments with respect to a set of real-valued functions f₁,…, f_m. Given a finite subset C⊆ S the algorithm produces a measure μ^* supported on C and we prove that μ^* is a good approximation to μ whenever the functions f₁,…, 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.

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