
On the robustness of noiseblind lowrank recovery from rankone measurements
We prove new results about the robustness of wellknown convex noisebli...
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On the convex geometry of blind deconvolution and matrix completion
Lowrank matrix recovery from structured measurements has been a topic o...
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Blind deconvolution of covariance matrix inverses for autoregressive processes
Matrix C can be blindly deconvoluted if there exist matrices A and B suc...
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SampleEfficient Low Rank Phase Retrieval
In this paper we obtain an improved guarantee for the Low Rank Phase Ret...
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Optimal Exploitation of Subspace Prior Information in Matrix Sensing
Matrix sensing is the problem of reconstructing a lowrank matrix from a...
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Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming
Statistical inference and information processing of highdimensional dat...
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Composite optimization for robust blind deconvolution
The blind deconvolution problem seeks to recover a pair of vectors from ...
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Phase Retrieval of LowRank Matrices by Anchored Regression
We study the lowrank phase retrieval problem, where we try to recover a d_1× d_2 lowrank matrix from a series of phaseless linear measurements. This is a fourthorder inverse problem, as we are trying to recover factors of matrix that have been put through a quadratic nonlinearity after being multiplied together. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope, and enforce the rank constraint through nuclear norm regularization. The result is a convex program that works in the space of d_1× d_2 matrices. We analyze two specific scenarios. In the first, the target matrix is rank1, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that the anchored regression program returns an accurate estimate from a nearoptimal number of measurements given that we have access to an anchor vector of sufficient quality. We also show how to create such an anchor in the phaseless blind deconvolution problem, again from an optimal number of measurements, and present a partial result in this direction for the general rank problem.
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