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PreprintCC BY 4.0arXiv

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(dlog2 d) when the rank r = O(1). We verify this convergence through numerical results.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026fusion,
  title = {{The Fusion Frame Phase Retrieval}},
  author = {Haixia Liu and Bing Gao and Yang Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08531},
  eprint = {2609.08531},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08531},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Fusion Frame Phase Retrieval
AU  - Haixia Liu
AU  - Bing Gao
AU  - Yang Wang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08531
UR  - https://arxiv.org/abs/2609.08531
AB  - 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(dlog^2 d) when the rank r = O(1). We verify this convergence through numerical results.
ER  -