Sample-Efficient Low Rank Phase Retrieval

This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an <inline-formula> <tex-math notation="LaTeX">n \times q </tex-math></inline-formula> rank-<inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formul...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 67; no. 12; pp. 8190 - 8206
Main Authors: Nayer, Seyedehsara, Vaswani, Namrata
Format: Journal Article
Language:English
Published: New York IEEE 01.12.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Complexity
Complexity theory
compressive sensing
Extraterrestrial measurements
Incoherence
low rank matrix recovery
Low-rank
Mathematical analysis
Matrix algebra
Noise measurement
Phase measurement
Phase retrieval
phase retrieval (PR)
Principal component analysis
Stability analysis
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an <inline-formula> <tex-math notation="LaTeX">n \times q </tex-math></inline-formula> rank-<inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> matrix <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">\boldsymbol {y}_{k} = | \boldsymbol {A}_{k}^\top \boldsymbol {x}^{\ast} _{k}| </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">k=1, 2,\ldots, q </tex-math></inline-formula>, when each <inline-formula> <tex-math notation="LaTeX">\boldsymbol {y}_{k} </tex-math></inline-formula> is an m-length vector containing independent phaseless linear projections of <inline-formula> <tex-math notation="LaTeX">\boldsymbol {x}^{\ast}_{k} </tex-math></inline-formula>. Here <inline-formula> <tex-math notation="LaTeX">|.| </tex-math></inline-formula> takes element-wise magnitudes of a vector. The different matrices <inline-formula> <tex-math notation="LaTeX">\boldsymbol {A}_{k} </tex-math></inline-formula> are i.i.d. and each contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was introduced and studied in our recent work. As long as the right singular vectors of <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> if the total number of measurements <inline-formula> <tex-math notation="LaTeX">mq \gtrsim nr^{2} (r + \log (1/\epsilon)) </tex-math></inline-formula>. In addition, we also need <inline-formula> <tex-math notation="LaTeX">m \gtrsim max(r, \log q, \log n) </tex-math></inline-formula> because of the specific asymmetric nature of our problem. Compared to our recent work, we improve the sample complexity of the AltMin iterations by a factor of <inline-formula> <tex-math notation="LaTeX">r^{2} </tex-math></inline-formula>, and that of the initialization by a factor of <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>. We argue, based on comparison with related well-studied problems, why the above sample complexity cannot be improved any further for non-convex solutions to LRPR. We also extend our result to the noisy case; we prove stability to corruption by small additive noise.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2021.3112805