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arXiv · 2609.08531

The Fusion Frame Phase Retrieval

Haixia Liu·Bing Gao·Yang Wang
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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.

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Haixia Liu, Bing Gao, Yang Wang. 2026-09-08. The Fusion Frame Phase Retrieval. https://arxiv.org/abs/2609.08531

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