TY - JOUR
T1 - Uniform Exact Reconstruction of Sparse Signals and Low-Rank Matrices From Phase-Only Measurements
AU - Chen, Junren
AU - Ng, Michael K.
N1 - The work of Junren Chen was supported by the Hong Kong Ph.D. Fellowship from the Hong Kong Research Grant Council. The work of Michael K. Ng was supported in part by the Hong Kong Research Grant Council GRF under Grant 12300218, Grant 12300519, Grant 17201020, Grant 17300021, Grant C1013-21GF, and Grant C7004-21GF; and in part by joint NSFC-RGC under Grant N-HKU76921.
PY - 2023/10
Y1 - 2023/10
N2 - In phase-only compressive sensing (PO-CS), our goal is to recover low-complexity signals (e.g., sparse signals, low-rank matrices) from the phase of complex linear measurements. While perfect recovery of signal direction in PO-CS was observed quite early, the exact reconstruction guarantee for a fixed, real signal was recently done by Jacques and Feuillen. However, two questions remain open: the uniform recovery guarantee and exact recovery of complex signal. In this paper, we almost completely address these two open questions. We prove that, all complex sparse signals or low-rank matrices can be uniformly, exactly recovered from a near optimal number of complex Gaussian measurement phases. By recasting PO-CS as a linear compressive sensing problem, the exact recovery follows from restricted isometry property (RIP). Our approach to uniform recovery guarantee is based on covering arguments that involve a delicate control of the (original linear) measurements with overly small magnitude. To work with complex signal, a different sign-product embedding property and a careful rescaling of the sensing matrix are employed. In addition, we show an extension that the uniform recovery is stable under moderate bounded noise. We also propose to add Gaussian dither before capturing the phases to achieve full reconstruction with norm information. Experimental results are reported to corroborate and demonstrate our theoretical results.
AB - In phase-only compressive sensing (PO-CS), our goal is to recover low-complexity signals (e.g., sparse signals, low-rank matrices) from the phase of complex linear measurements. While perfect recovery of signal direction in PO-CS was observed quite early, the exact reconstruction guarantee for a fixed, real signal was recently done by Jacques and Feuillen. However, two questions remain open: the uniform recovery guarantee and exact recovery of complex signal. In this paper, we almost completely address these two open questions. We prove that, all complex sparse signals or low-rank matrices can be uniformly, exactly recovered from a near optimal number of complex Gaussian measurement phases. By recasting PO-CS as a linear compressive sensing problem, the exact recovery follows from restricted isometry property (RIP). Our approach to uniform recovery guarantee is based on covering arguments that involve a delicate control of the (original linear) measurements with overly small magnitude. To work with complex signal, a different sign-product embedding property and a careful rescaling of the sensing matrix are employed. In addition, we show an extension that the uniform recovery is stable under moderate bounded noise. We also propose to add Gaussian dither before capturing the phases to achieve full reconstruction with norm information. Experimental results are reported to corroborate and demonstrate our theoretical results.
KW - Compressed sensing
KW - low-rankness
KW - phase-only measurement
KW - sparsity
KW - uniform recovery
UR - http://www.scopus.com/inward/record.url?scp=85164668672&partnerID=8YFLogxK
U2 - 10.1109/TIT.2023.3293830
DO - 10.1109/TIT.2023.3293830
M3 - Journal article
AN - SCOPUS:85164668672
SN - 0018-9448
VL - 69
SP - 6739
EP - 6764
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 10
ER -