Fusion frame approach enables faster phase retrieval with projections
The Fusion Frame Phase Retrieval
Information Theory
Summary
Phase retrieval is about figuring out a signal from measurements where only the size is known, not the direction. This paper studies a version where measurements come from special projection matrices instead of typical random ones. The authors develop new mathematical tools to show that a gradient descent method quickly finds the original signal with fewer measurements. They also confirm their approach works with computer experiments.
What this means in practice
- •For signal processing engineers: Use new measurement designs and algorithms to reconstruct signals more efficiently from magnitude-only data in imaging or communications.
- •For quantum information teams: Better understand and analyze measurements involving rank-r projections, aiding quantum state reconstruction tasks.
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.