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Phase Retrieval Without Small-Ball Probability Assumptions: Recovery Guarantees for PhaseLift



Yi-Kai Liu, Felix Krahmer


We study the problem of recovering an unknown vector x in R^n from measurements of the form y_i = |a_i^T x|^2 (for i=1,...,m), where the vectors a_i in R^n are chosen independently at random, with each coordinate a_{ij} being chosen independently from a fixed sub-Gaussian distribution D. However, without making additional assumptions on the random variables a_{ij} --- for example on the behavior of their small ball probabilities --- it may happen some vectors x cannot be uniquely recovered. We show that for any sub-Gaussian distribution D, with no additional assumptions, it is still possible to recover most vectors x. More precisely, one can recover those vectors x that are "not too peaky" in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The recovery guarantees in this paper are for the PhaseLift algorithm, a tractable convex program based on a matrix formulation of the problem. We prove uniform recovery of all "not too peaky" vectors from m = O(n) measurements, in the presence of noise. This extends previous work on PhaseLift by Candes and Li.
Arxiv preprint server


Phase retrieval, low-rank matrix recovery, convex relaxation, crystallography, optical imaging
Created September 28, 2017, Updated November 10, 2018