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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| Vydané v: | IEEE transactions on information theory Ročník 67; číslo 12; s. 8190 - 8206 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
IEEE
01.12.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | 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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| AbstractList | 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. This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an [Formula Omitted] rank-[Formula Omitted] matrix [Formula Omitted] from [Formula Omitted], [Formula Omitted], when each [Formula Omitted] is an m-length vector containing independent phaseless linear projections of [Formula Omitted]. Here [Formula Omitted] takes element-wise magnitudes of a vector. The different matrices [Formula Omitted] 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 [Formula Omitted] satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to [Formula Omitted] if the total number of measurements [Formula Omitted]. In addition, we also need [Formula Omitted] 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 [Formula Omitted], and that of the initialization by a factor of [Formula Omitted]. 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. |
| Author | Nayer, Seyedehsara Vaswani, Namrata |
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| Snippet | 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>... This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an [Formula Omitted] rank-[Formula Omitted] matrix [Formula Omitted] from [Formula... |
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| SubjectTerms | 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 |
| Title | Sample-Efficient Low Rank Phase Retrieval |
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