Universität zu Köln

Kliesch, Martin, Kueng, Richard, Eisert, Jens ORCID: 0000-0003-3033-1292 and Gross, David (2016). Improving Compressed Sensing With the Diamond Norm. IEEE Trans. Inf. Theory, 62 (12). S. 7445 - 7464. PISCATAWAY: IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC. ISSN 1557-9654

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Abstract

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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Kliesch, Martin UNSPECIFIED UNSPECIFIED UNSPECIFIED Kueng, Richard UNSPECIFIED UNSPECIFIED UNSPECIFIED Eisert, Jens UNSPECIFIED orcid.org/0000-0003-3033-1292 UNSPECIFIED Gross, David UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-253763
DOI:	10.1109/TIT.2016.2606500
Journal or Publication Title:	IEEE Trans. Inf. Theory
Volume:	62
Number:	12
Page Range:	S. 7445 - 7464
Date:	2016
Publisher:	IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
Place of Publication:	PISCATAWAY
ISSN:	1557-9654
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Physics > Institute for Theoretical Physics
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language CONVEX; MATRICES; MINIMIZATION; RECOVERY; SYSTEMS Multiple languages Computer Science, Information Systems; Engineering, Electrical & Electronic Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/25376