We tackle the problem of recovering a complex signal $\vx\in\mathbb{C}^n$ from quadratic measurements of the form $y_i=\vx^*\vA_i\vx$, where $\{\vA_i\}_{i=1}^m$ is a set of complex iid standard Gaussian matrices. This nonconvex problem is related to the well understood phase retrieval problem where $\vA_i$ is a rank1 positive semidefinite matrix. Here we study a general fullrank case which models a number of key applications such as molecular geometry recovery from distance distributions and compound measurements in phaseless diffractive imaging. Most prior work either addresses the rank1 case or focuses on real measurements. The several papers that address the fullrank complex case adopt the semidefinite relaxation approach and are thus computationally demanding. In this paper we propose a method based on the standard framework comprising a spectral initialization followed by iterative gradient descent updates. We prove that when the number of measurements exceeds the signal's length by some constant factor, a globally optimal solution can be recovered from complex quadratic measurements with high probability. Numerical experiments on simulated data corroborate our theoretical analysis.
Phase retrieval of lowrank matrices by anchored regression
Abstract We study the lowrank phase retrieval problem, where our goal is to recover a $d_1\times d_2$ lowrank matrix from a series of phaseless linear measurements. This is a fourthorder inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a nearoptimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show more »
 Award ID(s):
 1718771
 Publication Date:
 NSFPAR ID:
 10302676
 Journal Name:
 Information and Inference: A Journal of the IMA
 Volume:
 10
 Issue:
 1
 ISSN:
 20498772
 Sponsoring Org:
 National Science Foundation
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