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Improving Compressed Sensing with the Diamond Norm
Publikationstyp
Journal Article
Date Issued
2016-12
Sprache
English
Volume
62
Issue
12
Start Page
7445
End Page
7463
Article Number
7562508
Citation
IEEE Transactions on Information Theory 62 (12): 7562508 (2016-12)
Publisher DOI
Scopus ID
In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this paper, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that-for a class of matrices saturating a certain norm inequality-the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include signal analysis tasks, such as blind matrix deconvolution or the retrieval of certain unitary basis changes, as well as the quantum information problem of process tomography with random measurements. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this paper touches on an aspect of the notoriously difficult tensor completion problem.
Subjects
Compressed sensing
inverse problems
quantum mechanics
state estimation
system recovery