The generic crystallographic phase retrieval problem

2023/07/13 by Edidin, Dan, Suresh, Arun · 1 citation
#42A10 #94A12 #94A15 #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2307.06835

Abstract

In this paper we consider the problem of recovering a signal x ∈ ℝ^N from its power spectrum assuming that the signal is sparse with respect to a generic basis for ℝ^N. Our main result is that if the sparsity level is at most ∼ N/2 in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is ∼ N/4 then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in ℂ^N which extend earlier results of Wang and Xu \citearXiv:1310.0873.

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