Phase retrieval improves signal recovery from special projection measurements
The Fusion Frame Phase Retrieval
Information Theory
Summary
The problem of phase retrieval is about figuring out a signal when you only know the strength of certain measurements but not their exact details. Most past work focused on random measurements that behave like Gaussian distributions. This paper looks at a special kind of measurement called fusion frames, which use particular projection matrices. The authors provide new mathematical tools to analyze this problem and show that a method called gradient descent can quickly find the original signal with high accuracy. They also back up their findings with computer simulations.
phase retrievalgradient descentfusion frameorthogonal projectionHaar measureconcentration inequalitieslinear measurementsrank-r projectionsignal reconstructionmeasurement complexity
Authors
Haixia Liu, Bing Gao, Yang Wang
Abstract
The phase retrieval problem involves reconstructing a function or signal solely from the magnitude of linear measurements. Most theoretical analyses of phase retrieval algorithms rely on i.i.d. Gaussian random measurements or sub-Gaussian random measurements. In this paper, our focus is on the fusion frame phase retrieval problem, where the sampling matrices are i.i.d. rank-$r$ orthogonal projections drawn from the Haar measure. We present concentration inequalities for functions on the set of rank-$r$ orthogonal projection matrices. These inequalities are crucial for the theoretical analysis of the fusion frame phase retrieval problem. Based on these inequalities, we demonstrate that gradient descent, combined with a two-stage initialization, achieves linear convergence to the target signal up to a global phase with a measurement complexity of $O(d\log^2 d)$ when the rank $r = O(1)$. We verify this convergence through numerical results.