Generalized phase retrieval : measurement number, matrix recovery and beyond

2016/05/25 by Yang Wang, Wang, Yang, Zhiqiang Xu +1 · 11 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #15A63 #42C15 (Primary) #57N35 (Secondary) #Advanced Electron Microscopy Techniques and Applications #Advanced X-ray Imaging Techniques #Bilinear form #Combinatorics #FOS: Computer and information sciences #FOS: Mathematics #Fourier transform #Functional Analysis (math.FA) #Geometry #Information Theory (cs.IT) #Invertible matrix #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Orthogonal basis #Orthogonal matrix #Phase (matter) #Phase retrieval #Physics #Pure mathematics #Quadratic equation #Quantum mechanics #Rank (graph theory) #Seismic Imaging and Inversion Techniques #cs.IT #math.FA #math.IT #msc:15A63 #msc:42C15 #msc:57N35

paper · pdf · doi:10.48550/arxiv.1605.08034

published in arXiv (Cornell University) (Cornell University) · 31 pages, add Theorem 5.2, 5.3

openalex publication_date 2016/05/25 · arxiv created 2016/06/03 · arxiv updated 2016/06/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06

Abstract

In this paper, we develop a framework of generalized phase retrieval in which one aims to reconstruct a vector \mathbf x in \mathbb R^d or \mathbb C^d through quadratic samples \mathbf x^*A₁\mathbf x, …, \mathbf x^*A_N\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 C^{d× d}. Applying the results to phase retrieval, we show that generic d × d matrices A₁,…, A_N 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,…,A_N, 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.

Citations

• PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

Cited by

• Phase Retrievability of Super Operators and Measurements
• Bilinear Compressive Security
• Optimal Frames for Phase Retrieval from Edge Vectors of Optimal Polygons
• On the Performance of Amplitude-Based Models for Low-Rank Matrix Recovery
• Phase Retrieval in \mathbb C⁴ Requires Exactly Eleven Measurements
• Conjugate Phase Retrieval on \mathbb C^M by real vectors
• Sparse Signal Recovery From Quadratic Systems with Full-Rank Matrices
• The Condition Number in Phase Retrieval from Intensity Measurements
• Signal Recovery on Algebraic Varieties Using Linear Samples
• Phase retrieval and matrix sensing via benign and overparametrized nonconvex optimization
• Phase retrieval by hyperplanes

Related

• Almost Everywhere Generalized Phase Retrieval
• The minimal measurement number for low-rank matrices recovery
• Sample-Efficient Low Rank Phase Retrieval
• Provable Low Rank Phase Retrieval
• Phase Retrieval By Projections