Sublinear-Time Algorithms for Compressive Phase Retrieval

Yi Li, Vasileios Nakos

Abstract: In the problem of compressed phase retrieval, the goal is to reconstruct a sparse or approximately k-sparse vector x ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> given access to y = |Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> |, where |v| denotes the vector obtained from taking the absolute value of v ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> coordinate-wise. In this paper we present sublinear-time algorithms for a few for-each variants of the compressive phase retrieval problem which are akin to the variants considered for the classical compressive sensing problem in theoretical computer science. Our algorithms use pure combinatorial techniques and near-optimal number of measurements.