2016/05/25 by Yang Wang, Wang, Yang, Zhiqiang Xu +1 · 7 citations
Biochemistry, Genetics and Molecular Biology · Earth and Planetary Sciences · Physics and Astronomy · #15A63 #42C15 (Primary) #57N35 (Secondary) #Advanced Electron Microscopy Techniques and Applications #Advanced X-ray Imaging Techniques #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Information Theory (cs.IT) #Seismic Imaging and Inversion Techniques
paper · pdf · doi:10.48550/arxiv.1605.08034
openalex publication_date 2016/05/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper, we develop a framework of generalized phase retrieval in which one aims to reconstruct a vector \mathbf x in \mathbb Rd or \mathbb Cd through quadratic samples \mathbf x^*A1\mathbf x, …, \mathbf x^*AN\mathbf x. 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∈ \mathbb 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. Our method also leads to a novel proof for the classical Stiefel-Hopf condition on nonsingular bilinear form. 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.