Generalized phase retrieval: Measurement number, matrix recovery and beyond

• Published in: Applied and computational harmonic analysis Vol. 47; no. 2; pp. 423 - 446
• Main Authors: Wang, Yang, Xu, Zhiqiang
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2019
• Subjects: Bilinear form; Embedding; Fourier transform; Frames; Fusion frames; Low rank matrix recovery; Measurement number; Phase retrieval; secondary; Frames; Fusion frames; Phase retrieval; Fourier transform; Embedding; Measurement number; Bilinear form; Low rank matrix recovery; primary
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2017.09.003

In this paper, we develop a framework of generalized phase retrieval in which one aims to reconstruct a vector x in Rd or Cd through quadratic samples x⁎A1x,…,x⁎ANx. The generalized phase retrieval includes as special cases the standard phase retrieval as well as the phase retrieval by orthogonal projections. We first explore the connections among generalized phase retrieval, low-rank matrix recovery and nonsingular bilinear form. Motivated by the connections, we present results on the minimal measurement number needed for recovering a matrix that lies in a set W∈Cd×d. Applying the results to phase retrieval, we show that generic d×d matrices A1,…,AN have the phase retrieval property if N≥2d−1 in the real case and N≥4d−4 in the complex case for very general classes of A1,…,AN, e.g. matrices with prescribed ranks or orthogonal projections. We also give lower bounds on the minimal measurement number required for generalized phase retrieval. For several classes of dimensions d we obtain the precise values of the minimal measurement number. Our work unifies and enhances results from the standard phase retrieval, phase retrieval by projections and low-rank matrix recovery.