eprintid: 2474
rev_number: 9
eprint_status: archive
userid: 6
dir: disk0/00/00/24/74
datestamp: 2015-01-13 14:18:27
lastmod: 2015-01-13 14:18:27
status_changed: 2015-01-13 14:18:27
type: article
metadata_visibility: show
creators_name: Piga, Dario
creators_name: Tóth, Roland
creators_id: dario.piga@imtlucca.it
creators_id: 
title: An SDP approach for l0-minimization: application to ARX model segmentation 
ispublished: pub
subjects: QA75
divisions: CSA
full_text_status: public
keywords: Compressive sensing; ℓ0ℓ0-minimization; Regularization; SDP relaxation; Sparse estimation; Segmentation
abstract: Abstract Minimizing the ℓ 0 -seminorm of a vector under convex constraints is a combinatorial (NP-hard) problem. Replacement of the ℓ 0 -seminorm with the ℓ 1 -norm is a commonly used approach to compute an approximate solution of the original ℓ 0 -minimization problem by means of convex programming. In the theory of compressive sensing, the condition that the sensing matrix satisfies the Restricted Isometry Property (RIP) is a sufficient condition to guarantee that the solution of the ℓ 1 -approximated problem is equal to the solution of the original ℓ 0 -minimization problem. However, the evaluation of the conservativeness of the ℓ 1 -relaxation approaches is recognized to be a difficult task in case the {RIP} is not satisfied. In this paper, we present an alternative approach to minimize the ℓ 0 -norm of a vector under given constraints. In particular, we show that an ℓ 0 -minimization problem can be relaxed into a sequence of semidefinite programming problems, whose solutions are guaranteed to converge to the optimizer (if unique) of the original combinatorial problem also in case the {RIP} is not satisfied. Segmentation of {ARX} models is then discussed in order to show, through a relevant problem in system identification, that the proposed approach outperforms the ℓ 1 -based relaxation in detecting piece-wise constant parameter changes in the estimated model. 
date: 2013-12
date_type: published
publication: Automatica
volume: 49
number: 12
publisher: Elsevier 
pagerange: 3646 - 3653
id_number: 10.1016/j.automatica.2013.09.021
refereed: TRUE
issn: 0005-1098
official_url: http://www.sciencedirect.com/science/article/pii/S0005109813004548
funders: This work was supported by the Netherlands Organization for Scientific Research (NWO, grant. no.: 639.021.127) and by the French ministries of Foreign Affairs, Education and Research and the French-Dutch Academy (PHC Van Gogh project, n. 29342QL)
citation:   Piga, Dario and Tóth, Roland  An SDP approach for l0-minimization: application to ARX model segmentation.  Automatica, 49 (12).  3646 - 3653.  ISSN 0005-1098      (2013)  
document_url: http://eprints.imtlucca.it/2474/1/Aut2013PigaPreprint2.pdf