The minimal measurement number for low-rank matrix recovery

Xu, Zhiqiang

doi:10.1016/j.acha.2017.01.005

Cited by 36 publications

(26 citation statements)

References 22 publications

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“…It was also conjectured in [17] that N = 4dr − 4r 2 is the minimal N for which there exists A = (A j ) N j=1 so that M A is injective on M d,r (F). In [39], the author proved the conjecture for F = C and disproved it for F = R, showing the being nonsingular. This connection has led us to also study nonsingular bilinear form, an area with deep historical roots.…”

Section: Phaseliftmentioning

confidence: 99%

“…For example for fusion frame phase retrieval in R d , it is known that a generic choice of N = 2d − 1 orthogonal projections A = (P j ) N j=1 with 0 < rank(P j ) < d has the phase retrieval property [7,16], but the smallest such N remains unknown in general. For d = 4, it is known that there exists a fusion frame A = (P j ) N j=1 with N = 6 = 2d − 2 [39] having the phase retrieval property. In this paper, we shall show the number N = 6 is tight for d = 4.…”

Section: Phaseliftmentioning

confidence: 99%

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Generalized phase retrieval: Measurement number, matrix recovery and beyond

Wang

2019

Applied and Computational Harmonic Analysis

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In this paper, we develop a framework of generalized phase retrieval in which one aims to reconstruct a vector x in R d or C d through quadratic samples x * A1x, . . . , x * AN x. The generalized phase retrieval includes as special cases the standard phase retrieval as well as the phase retrieval by orthogonal projections. We first explore the connections among generalized phase retrieval, low-rank matrix recovery and nonsingular bilinear form. Motivated by the connections, we present results on the minimal measurement number needed for recovering a matrix that lies in a set W ∈ C d×d . Applying the results to phase retrieval, we show that generic d × d matrices A1, . . . , AN have the phase retrieval property if N ≥ 2d − 1 in the real case and N ≥ 4d − 4 in the complex case for very general classes of A1, . . . , AN , e.g. matrices with prescribed ranks or orthogonal projections. Our method also leads to a novel proof for the classical Stiefel-Hopf condition on nonsingular bilinear form. We also give lower bounds on the minimal measurement number required for generalized phase retrieval. For several classes of dimensions d we obtain the precise values of the minimal measurement number. Our work unifies and enhances results from the standard phase retrieval, phase retrieval by projections and low-rank matrix recovery.2010 Mathematics Subject Classification. Primary 42C15, Secondary 94A12, 15A63, 15A83 .

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Section: Phaseliftmentioning

confidence: 99%

Section: Phaseliftmentioning

confidence: 99%

Generalized phase retrieval: Measurement number, matrix recovery and beyond

Wang

2019

Applied and Computational Harmonic Analysis

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“…However we obtain the following necessity result. This result was independently obtained by Zhiqiang Xu in his recent paper [10]. Theorem 1.6.…”

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confidence: 73%

Projections and phase retrieval

Edidin

2017

Applied and Computational Harmonic Analysis

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Abstract. We characterize collections of orthogonal projections for which it is possible to reconstruct a vector from the magnitudes of the corresponding projections. As a result we are able to show that in an M -dimensional real vector space a vector can be reconstructed from the magnitudes of its projections onto a generic collection of N ≥ 2M − 1 subspaces. We also show that this bound is sharp when N = 2 k + 1. The results of this paper answer a number of questions raised in [4].

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“…Evidently as long as A is injective on the set of matrices of rank at most r, X will be the unique solution to (3). Indeed, it has been shown that if A consists of m ě 4nr´4r 2 generic measurement matrices, then A will be injective; see [14,106] for more details. Despite this, the rank minimization problem is known to be NP-hard and computationally intractable since it is an extension of the 0 -minimization problem in compressed sensing [42,23].…”

Section: Convex Approach: Nuclear Norm Minimizationmentioning

confidence: 99%

Exploiting the Structure Effectively and Efficiently in Low-Rank Matrix Recovery

Cai

2018

Handbook of Numerical Analysis

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Low rank model arises from a wide range of applications, including machine learning, signal processing, computer algebra, computer vision, and imaging science. Low rank matrix recovery is about reconstructing a low rank matrix from incomplete measurements. In this survey we review recent developments on low rank matrix recovery, focusing on three typical scenarios: matrix sensing, matrix completion and phase retrieval. An overview of effective and efficient approaches for the problem is given, including nuclear norm minimization, projected gradient descent based on matrix factorization, and Riemannian optimization based on the embedded manifold of low rank matrices. Numerical recipes of different approaches are emphasized while accompanied by the corresponding theoretical recovery guarantees. This is an ill-posed problem without assuming any structure on X since there are more unknowns than equations. However, noticing that the number of degrees of freedom in an n by n rank r matrix is p2n´rqr [100] which can be much smaller than n 2 provided r is small, it is reasonable to expect to reconstruct a low rank matrix from fewer than n 2 measurements.Authors are listed alphabetically.

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The minimal measurement number for low-rank matrix recovery

Cited by 36 publications

References 22 publications

Generalized phase retrieval: Measurement number, matrix recovery and beyond

Generalized phase retrieval: Measurement number, matrix recovery and beyond

Projections and phase retrieval

Exploiting the Structure Effectively and Efficiently in Low-Rank Matrix Recovery

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