Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming

Abstract

Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, 1 -minimization, guarantees recovery of sparse parameter signals even when the system is underdetermined. In this paper, we consider a more challenging problem: when the phase of the output measurements from a linear system is omitted. Using a lifting technique, we show that even though the phase information is missing, the sparse signal can be recovered exactly by solving a simple semidefinite program when the sampling rate is sufficiently high, albeit the exact solutions to both sparse signal recovery and phase retrieval are combinatorial. The results extend the type of applications that compressive sensing can be applied to those where only output magnitudes can be observed. We demonstrate the accuracy of the algorithms through theoretical analysis, extensive simulations and a practical experiment.

PhaseCode: Fast and Efficient Compressive Phase Retrieval Based on Sparse-Graph Codes

“…This number was later improved to 4K − 2 in [31], [32]. The PhaseLift method is also proposed for the sparse case in [14] and [17], requiring Θ(K 2 log(n)) intensity measurements, and having a computational complexity of O(n 3 ), making the method less practical for large-scale applications. In [33], the authors propose an efficient algorithm based on polarization method that is able to stably reconstruct any K-sparse vector from Θ(K log(n)) noisy intensity measurements with complexity polynomial in n. The alternating minimization method in [16] can also be adapted to the sparse case with Θ(K 2 log(n)) measurements and a complexity of O(K 3 n log(n)).…”

“…Indeed, the problem of recovering a signal from only the magnitude of its Fourier transform has been a well-studied problem in the signal processing literature for several decades under the umbrella of phase retrieval [11]. It has recently received renewed interest in the "postcompressed-sensing" era [12]- [14], allowing for the insights from compressive sensing to be incorporated into the phase retrieval problem when the signal of interest is sparse, and the measurement matrix is unconstrained.…”

Vectorial Phase Retrieval of 1-D Signals

“…Hence, they may be decomposed as follows, (30) Similarly, the FT of , can also be factored as (31) where as in the proof of Thm. 1, .…”

“…The problem is to recover the signal from modulus measurements of many dot products . As shown in [29], [31], this problem can be solved by convex programming via a low rank matrix completion formulation. As proven in [32], with high probability, this formulation is robust to measurement noise, and in fact (w.h.p.)…”

Fast compressive phase retrieval from Fourier measurements

“…However, to the best of our knowledge there is no algorithm approaching this bound presently. Using PhaseLift [10], Ohlsson et al [11] proposed an recovery algorithm from O(k 2 log N ) measurements. However, this technique is based on semidefinite programming and suffers from high computational complexity.…”