Measurement Matrix Design for Phase Retrieval Based on Mutual Information

Abstract

In phase retrieval problems, a signal of interest (SOI) is reconstructed based on the magnitude of a linear transformation of the SOI observed with additive noise. The linear transform is typically referred to as a measurement matrix. Many works on phase retrieval assume that the measurement matrix is a random Gaussian matrix, which, in the noiseless scenario with sufficiently many measurements, guarantees invertability of the transformation between the SOI and the observations, up to an inherent phase ambiguity. However, in many practical applications, the measurement matrix corresponds to an underlying physical setup, and is therefore deterministic, possibly with structural constraints. In this work we study the design of deterministic measurement matrices, based on maximizing the mutual information between the SOI and the observations. We characterize necessary conditions for the optimality of a measurement matrix, and analytically obtain the optimal matrix in the low signal-to-noise ratio regime. Practical methods for designing general measurement matrices and masked Fourier measurements are proposed. Simulation tests demonstrate the performance gain achieved by the suggested techniques compared to random Gaussian measurements for various phase recovery algorithms.

Dynamic Metasurface Antennas for Uplink Massive MIMO Systems

“…Since finding the constrained matrix Q which maximizes (9) is a difficult task, we propose to set Q to be the closest feasible matrix to the unconstrained Q OD in the sense of minimal Frobenious norm. Here, as in [10], [55], we exploit the fact that R OD s is invariant to the selection of the left singular matrix Ũ and the diagonal singular values matrix D, and set these matrices such that the Frobenious distance to the feasible approximation is minimized. To formulate the problem, we let Q K×N be the set of K × N which can be written as in (3) and whose non-zero entries belong to the feasible set Q.…”

Unfolded Algorithms for Deep Phase Retrieval

“…In particular, a system design perspective for a phase retrieval data acquisition system can be defined as designing the sensing matrix in a deterministic fashion according to a performance criterion. As an example, [53] has considered the design of the sensing matrix in a deterministic manner such that it maximizes the mutual information between the signal of interest and the acquired phaseless measurements. However, the development of effective signal reconstruction techniques that can harness this maximal mutual information obtained by a judicious design of the sensing matrix is still an open problem.…”

“…The interest in learning techniques such as convolutional neural nets and stacked denoised autoencoders [49], [50], [52] has increased greatly. This has led to works such as [53], where the measurement matrix is designed based on mutual information, but the ideas are not exploited for recovery. A deep architecture that could handle the design of the sensing matrix along with the recovery task would be able improve the accuracy and efficiency of phase retrieval in a way unmatched by previous methods.…”

Maximum Entropy Low-Rank Matrix Recovery

“…In other words, the information-greedy sequential design (23) chooses the mask which maximizes the conditional variance of an observation, given data y n and current subspace estimates (Û n ,V n ). In the special case where (a) (PÛ n , PV n ) are the true subspaces (P U , P V ), and (b) all prior masks A 1:n are uncorrelated, (25) reduces to the PCA construction (24). Outside of this, (25) offers a generalization of (24) with two key advantages for active matrix recovery.…”

“…This includes the seminal paper [20] (see also [21]), who showed the profound fact that, for linear vector Gaussian channels, the gradient of the mutual information is related to the minimum mean-squared error matrix for parameter estimation. Such a result is further developed by [22], [23] and [24] for designing measurement matrices in compressive sensing and phase retrieval. The key novelty in our work is that, instead of directly maximizing the mutual information between signal (i.e., X) and measurements (i.e., y), we examine a dual (but equivalent) problem of maximizing the entropy of observations y.…”