Capacity-approaching PhaseCode for low-complexity compressive phase retrieval

Ramtin Pedarsani, Kangwook Lee, Kannan Ramchandran

Abstract: In this paper, we tackle the general compressive phase retrieval problem. The problem is to recover (to within a global phase uncertainty) a K-sparse complex vector of length n, x ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , from the magnitudes of m linear measurements, y = |Ax|, where A ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m×n</sup> can be designed, and the magnitudes are taken component-wise for vector Ax ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> . We propose a variant of the PhaseCode algorithm, first introduced in [1], and show that under some mild assumptions, using an irregular left-degree sparse-graph code construction, the algorithm can recover almost all the K non-zero signal components using only slightly more than 4K measurements, with orderoptimal time and memory complexity of O(K). It is known that the fundamental limit for the number of measurements in compressive phase retrieval problem is 4K - o(K) [2, 3]. To the best of our knowledge, this is the first constructive capacityapproaching compressive phase retrieval algorithm: in fact, our algorithm is also order-optimal in complexity and memory.