TY - JOUR
T1 - Support Recovery for Sparse Multidimensional Phase Retrieval
AU - Novikov, Alexei
AU - White, Stephen
N1 - Funding Information:
Manuscript received October 26, 2020; revised May 10, 2021; accepted July 9, 2021. Date of publication July 29, 2021; date of current version August 11, 2021. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Yuejie Chi. This work was supported by NSF under Grants DMS-1813943 and AFOSR FA9550-20-1-0026. (Corresponding author: Stephen White.) The authors are with the Department of Mathematics, Penn State University, University Park, PA 16802 USA (e-mail: novikov@psu.edu; sew347@psu.edu).
Funding Information:
This work was supported by NSF under Grants DMS-1813943 andAFOSRFA9550-20-1-0026.
Publisher Copyright:
© 1991-2012 IEEE.
PY - 2021
Y1 - 2021
N2 - We consider the sparse phase retrieval problem of recovering a sparse signal x in R^d from intensity-only measurements in dimension d ≥ 2. Sparse phase retrieval can often be equivalently formulated as the problem of recovering a signal from its autocorrelation, which is in turn directly related to the combinatorial problem of recovering a set from its pairwise differences. In one spatial dimension, this problem is well studied and known as the turnpike problem. In this work, we present MISTR (Multidimensional Intersection Sparse supporT Recovery), an algorithm which exploits this formulation to recover the support of a multidimensional signal from magnitude-only measurements. MISTR takes advantage of the structure of multiple dimensions to provably achieve the same accuracy as the best one-dimensional algorithms in dramatically less time. We prove theoretically that MISTR correctly recovers the support of signals distributed as a Gaussian point process with high probability as long as sparsity is at most O(ndθ ) for any θ < 1/2, where n^d represents pixel size in a fixed image window. In the case that magnitude measurements are corrupted by noise, we provide a thresholding scheme with theoretical guarantees for sparsity at most O(ndθ ) for θ < 1/4 that obviates the need for MISTR to explicitly handle noisy autocorrelation data. Detailed and reproducible numerical experiments demonstrate the effectiveness of our algorithm, showing that in practice MISTR enjoys time complexity which is nearly linear in the size of the input.
AB - We consider the sparse phase retrieval problem of recovering a sparse signal x in R^d from intensity-only measurements in dimension d ≥ 2. Sparse phase retrieval can often be equivalently formulated as the problem of recovering a signal from its autocorrelation, which is in turn directly related to the combinatorial problem of recovering a set from its pairwise differences. In one spatial dimension, this problem is well studied and known as the turnpike problem. In this work, we present MISTR (Multidimensional Intersection Sparse supporT Recovery), an algorithm which exploits this formulation to recover the support of a multidimensional signal from magnitude-only measurements. MISTR takes advantage of the structure of multiple dimensions to provably achieve the same accuracy as the best one-dimensional algorithms in dramatically less time. We prove theoretically that MISTR correctly recovers the support of signals distributed as a Gaussian point process with high probability as long as sparsity is at most O(ndθ ) for any θ < 1/2, where n^d represents pixel size in a fixed image window. In the case that magnitude measurements are corrupted by noise, we provide a thresholding scheme with theoretical guarantees for sparsity at most O(ndθ ) for θ < 1/4 that obviates the need for MISTR to explicitly handle noisy autocorrelation data. Detailed and reproducible numerical experiments demonstrate the effectiveness of our algorithm, showing that in practice MISTR enjoys time complexity which is nearly linear in the size of the input.
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U2 - 10.1109/TSP.2021.3096796
DO - 10.1109/TSP.2021.3096796
M3 - Article
AN - SCOPUS:85112674763
VL - 69
SP - 4403
EP - 4415
JO - IRE Transactions on Audio
JF - IRE Transactions on Audio
SN - 1053-587X
M1 - 9502003
ER -