Phase retrieval and norm retrieval

Bahmanpour, Saeid; Cahill, Jameson; Casazza, Peter G.; Jasper, John; Woodland, Lindsey M.

doi:10.48550/arxiv.1409.8266

Cited by 10 publications

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“…Note that the generalized phase retrieval problem includes the standard phase retrieval problem as a special case, with the additional restrictions A j 0 and rank(A j ) = 1. It also includes the so-called fusion frame (or projection) phase retrieval as a special case where each A j is an orthogonal projection matrix, namely A 2 j = A j [16,7,1]. Moreover, it is very closely related to and a generalization of the problem of information completeness of positive operator valued measures (POVMs) with respect to pure states in quantum tomography [21], where the norm of the vector we try to recover x ∈ C d is assumed to be 1.…”

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confidence: 99%

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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confidence: 99%

Generalized phase retrieval: Measurement number, matrix recovery and beyond

Wang

2019

Applied and Computational Harmonic Analysis

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“…, u j,d j } ⊂ R n and P j : R n → W j is an orthogonal projection. Following [1,5], phase retrieval by projection is to recover x ∈ R n up to a unimodular constant from P j x 2 . We say that {W j } m j=1 yields phase retrieval if for all x, y ∈ R n satisfying P j x = P j y for all j = 1, .…”

Section: Setmentioning

confidence: 99%

“…Particularly, they showed that phase retrieval can be done in R n with 2n − 1 subspaces each of any dimension less than n − 1. A question is also raised in [1] which states can phase retrieval be done in R n with fewer than 2n − 1 projections? Using the results in this paper, we present a positive answer for the question provided n is in the form of 2 k + 1.…”

Section: Setmentioning

confidence: 99%

“…The following theorem shows that 2n − 1 projections are enough for phase retrieval by projection. The problem is also raised in [1] which states can phase retrieval be done in R n with fewer than 2n − 1 projections? Based on Theorem 3.3, we show that the bound 2n − 1 is tight provided n = 2 k + 1, k ∈ Z ≥1 .…”

Section: Phase Retrieval By Projectionsmentioning

confidence: 99%

“…This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than 2n − 1 orthogonal projections are possible for the recovery of x ∈ R n from the norm of them, which gives a negative answer for a question raised in [1].…”

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See 2 more Smart Citations

The minimal measurement number for low-rank matrix recovery

2018

Applied and Computational Harmonic Analysis

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Signal Reconstruction From the Magnitude of Subspace Components

Bachoc

Ehler²

2015

IEEE Trans. Inform. Theory

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Abstract-We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of p-fusion frames and list decoding, we propose an algorithm that outputs a finite list of candidate signals, one of which is the correct one. In the random setting, we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program.

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Phase retrieval and norm retrieval

Cited by 10 publications

References 0 publications

Generalized phase retrieval: Measurement number, matrix recovery and beyond

Generalized phase retrieval: Measurement number, matrix recovery and beyond

The minimal measurement number for low-rank matrix recovery

Signal Reconstruction From the Magnitude of Subspace Components

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