Phase Retrieval with Polarization

Alexeev, Boris V.; Bandeira, Afonso S.; Fickus, Matthew; Mixon, Dustin G.

doi:10.1137/12089939x

Cited by 131 publications

(181 citation statements)

References 57 publications

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“…Let Γ = {γ k , 1 ≤ k ≤ a + b + 1} be the (unique) biorthogonal system to Γ . Let T be an invertible operator that maps an orthonormal set ∆ = {δ 1 …”

Section: New Analysis Resultsmentioning

confidence: 99%

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Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Bălan

2015

Found Comput Math

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Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well as an iterative algorithm that finds the least-square solution which is robust in the presence of noise. We analyze its numerical performance by comparing it to the Cramer-Rao lower bound.

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“…Let Γ = {γ k , 1 ≤ k ≤ a + b + 1} be the (unique) biorthogonal system to Γ . Let T be an invertible operator that maps an orthonormal set ∆ = {δ 1 …”

Section: New Analysis Resultsmentioning

confidence: 99%

“…The authors propose a novel algorithm adapted to compactly supported signals and FFT that uses individual signal spectral powers and two additional interferences between signals. A 3-term polarization identity has been used in [1] together with the angular synchronization algorithm.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Bălan

2015

Found Comput Math

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“…It is also closely related to the phase-retrieval problem in signal processing (20)(21)(22). An important qualitative difference with respect to the previous example (Z 2 synchronization) lies in the fact that Uð1Þ is a continuous group.…”

Section: Significancementioning

confidence: 97%

Phase transitions in semidefinite relaxations

Javanmard

Montanari

Ricci-Tersenghi

2016

Proc. Natl. Acad. Sci. U.S.A.

107

125

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Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large-scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that when the statistical noise is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several detection thresholds, as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems. Modern datasets pose new challenges to this centuries-old framework. On one hand, high-dimensional applications require the simultaneous estimation of millions of parameters. Examples span genomics (2), imaging (3), web services (4), and so on. On the other hand, the unknown object to be estimated has often a combinatorial structure: In clustering we aim at estimating a partition of the data points (5). Network analysis tasks usually require identification of a discrete subset of nodes in a graph (6, 7). Parsimonious data explanations are sought by imposing combinatorial sparsity constraints (8).There is an obvious tension between the above requirements. Although efficient algorithms are needed to estimate a large number of parameters, the maximum likelihood (ML) method often requires the solution of NP-hard (nondeterministic polynomial-time hard) combinatorial problems. A flourishing line of work addresses this conundrum by designing effective convex relaxations of these combinatorial problems (9-11).Unfortunately, the statistical properties of such convex relaxations are well understood only in a few cases [compressed sensing being the most important success story (12-14)]. In this paper we use tools from statistical mechanics to develop a precise picture of the behavior of a class of semidefinite programming relaxations. Relaxations of this type appear to be surprisingly effective in a variety of problems ranging from clustering to graph synchronization. For the sake of concreteness we will focus on three specific problems. Z 2 SynchronizationIn the general synchronization problem, we aim at estimating x 0,1 , x 0,2 , . . . , x 0,n , which are unknown elements of ...

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“…independent Gaussian random vectors, then m ≥ O(n) measurements suffice for injectivity of the map x → | a i , x | 2 m i=1 up to phase [4,19]. There is still a question of how to perform the inverse map in an efficient and stable manner, and in recent years several different algorithms have been proposed in this direction, see for example [1][2][3]8,9,12,14,19,25]. In particular, [3] noted that such measurements may be reformulated as y i = tr (a i a * i xx * ) , so that one can consider this problem as that of recovering an unknown rank-one positive semi-definite matrix.…”

Section: Introductionmentioning

confidence: 99%

The Local Convexity of Solving Systems of Quadratic Equations

2016

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This paper considers the recovery of a rank r positive semidefinite matrix XX T ∈ R n×n from m scalar measurements of the form yi := a T i XX T ai (i.e., quadratic measurements of X). Such problems arise in a variety of applications, including covariance sketching of highdimensional data streams, quadratic regression, quantum state tomography, among others. A natural approach to this problem is to minimize the loss function f (U ) = i (yi − a T i UU T ai) 2 which has an entire manifold of solutions given by {XO}O∈O r where Or is the orthogonal group of r × r orthogonal matrices; this is non-convex in the n × r matrix U , but methods like gradient descent are simple and easy to implement (as compared to semidefinite relaxation approaches). In this paper we show that once we have m ≥ Cnr log 2 (n) samples from isotropic gaussian ai, with high probability (a) this function admits a dimension-independent region of local strong convexity on lines perpendicular to the solution manifold, and (b) with an additional polynomial factor of r samples, a simple spectral initialization will land within the region of convexity with high probability. Together, this implies that gradient descent with initialization (but no re-sampling) will converge linearly to the correct X, up to an orthogonal transformation. We believe that this general technique (local convexity reachable by spectral initialization) should prove applicable to a broader class of nonconvex optimization problems.Mathematics Subject Classification. 65K99.

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Phase Retrieval with Polarization

Cited by 131 publications

References 57 publications

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Phase transitions in semidefinite relaxations

The Local Convexity of Solving Systems of Quadratic Equations

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