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A Class of Nonconvex Penalties Preserving OverallConvexity in Optimization-Based Mean Filtering
KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.ORCID iD: 0000-0003-0355-2663
KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.ORCID iD: 0000-0002-1927-1690
2016 (English)In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 65, no 24, 6650-6664 p.Article in journal (Refereed) Published
Abstract [en]

l1 mean filtering is a conventional, optimizationbasedmethod to estimate the positions of jumps in a piecewiseconstant signal perturbed by additive noise. In this method, the l1 norm penalizes sparsity of the first-order derivative of the signal.Theoretical results, however, show that in some situations, whichcan occur frequently in practice, even when the jump amplitudes tend to , the conventional method identifies false change points.This issue is referred to as stair-casing problem in this paper andrestricts practical importance of l1 mean filtering. In this paper, sparsity is penalized more tightly than the l1 norm by exploiting a certain class of nonconvex functions, while the strict convexity ofthe consequent optimization problem is preserved. This results in a higher performance in detecting change points. To theoretically justify the performance improvements over l1 mean filtering, deterministic and stochastic sufficient conditions for exact changepoint recovery are derived. In particular, theoretical results show that in the stair-casing problem, our approach might be able to exclude the false change points, while l1 mean filtering may fail. A number of numerical simulations assist to show superiorityof our method over l1 mean filtering and another state-of-theart algorithm that promotes sparsity tighter than the l1 norm. Specifically, it is shown that our approach can consistently detectchange points when the jump amplitudes become sufficiently large, while the two other competitors cannot.

Place, publisher, year, edition, pages
Institute of Electrical and Electronics Engineers (IEEE), 2016. Vol. 65, no 24, 6650-6664 p.
Keyword [en]
Change point recovery, mean filtering, nonconvex penalty, piecewise constant signal, sparse signal processing, total variation denoising
National Category
Signal Processing
Identifiers
URN: urn:nbn:se:kth:diva-195838DOI: 10.1109/TSP.2016.2612179ISI: 000386445000022ScopusID: 2-s2.0-84994013276OAI: oai:DiVA.org:kth-195838DiVA: diva2:1045590
Funder
Swedish Research Council
Note

QC 20161121

Available from: 2016-11-10 Created: 2016-11-10 Last updated: 2017-01-17Bibliographically approved

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Malek Mohammadi, MohammadrezaRojas, CristianWahlberg, Bo
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