Using Black-box Compression Algorithms for Phase Retrieval

Abstract

Compressive phase retrieval refers to the problem of recovering a structured n-dimensional complex-valued vector from its phase-less under-determined linear measurements. The non-linearity of measurements makes designing theoretically-analyzable efficient phase retrieval algorithms challenging. As a result, to a great extent, algorithms designed in this area are developed to take advantage of simple structures such as sparsity and its convex generalizations. The goal of this paper is to move beyond simple models through employing compression codes. Such codes are typically developed to take advantage of complex signal models to represent the signals as efficiently as possible.In this work, it is shown how an existing compression code can be treated as a black box and integrated into an efficient solution for phase retrieval. First, COmpressive PhasE Retrieval (COPER) optimization, a computationallyintensive compression-based phase retrieval method, is proposed. COPER provides a theoretical framework for studying compression-based phase retrieval. The number of measurements required by COPER is connected to κ, the α-dimension (closely related to the rate-distortion dimension) of the given family of compression codes. To finds the solution of COPER, an efficient iterative algorithm called gradient descent for COPER (GD-COPER) is proposed.It is proven that under some mild conditions on the initialization, if the number of measurements is larger than Cκ 2 log 2 n, where C is a constant, GD-COPER obtains an accurate estimate of the input vector in polynomial time.In the simulation results, JPEG2000 is integrated in GD-COPER to confirm the superb performance of the resulting algorithm on real-world images. I. INTRODUCTION A. MotivationConsider the problem of recovering x ∈ Q from m noisy phase-less linear observationswhere A ∈ C m×n and ∈ R m denote the sensing matrix and the measurement noise, respectively. Here Q denotes a compact subset of C n and | • | denotes the element-wise absolute value operator. Assume that the class of signals denotes by Q is "structured", but instead of the set Q, or its underlying structure, for recovering x from y, we have access to a compression code that takes advantage of the structure of signals in Q to compress them efficiently. For instance, consider the class of images or videos for which compression algorithms, such as JPEG2000 or MPEG4, capture complicated structures within such signals and encode them efficiently. Employing such structures in a phase retrieval algorithm can reduce the number of measurements or equivalently increase the quality of the recovered signals. This raises the following questions:

Universality of Linearized Message Passing for Phase Retrieval with Structured Sensing Matrices

“…This algorithm achieves non-trivial (better than random) performance when κ < 2/3 and exact recovery when κ < 0.63 [49]. While our analysis currently doesn't cover the non-linear iteration (7), we hope our techniques can be extended to analyze (7).…”

“…A representative, but necessarily incomplete list of such works includes the analysis of convex relaxations like PhaseLift due to Candès et al [22], Candès and Li [21] and PhaseMax due to Bahmani and Romberg [6], Goldstein and Studer [36] and analysis of non-convex optimization based methods due to Netrapalli et al [55], Candès et al [25] and Sun et al [65]. The number of measurements required if the underlying signal has a low dimensional structure has also been investigated [16,7,42].…”