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Axioms 2013, 2(3), 311-344; doi:10.3390/axioms2030311

Complexity L0-Penalized M-Estimation: Consistency in More Dimensions

1
Institute of Computational Biology, German Research Center for Environmental Health, Ingolstädter Landstr. 1, Neuherberg D-85764, Germany
2
Computer Systems Institute, Swiss Federal Institute of Technology (ETH), Zürich 8092, Switzerland
3
Department of Mathematics and Computer Science, University of Greifswald, Domstr. 11, Greifswald 17489, Germany
4
Mathematical Institute, Ludwig-Maximilian University of Munich Professor-Huber-Platz 2, München 80539, Germany
*
Author to whom correspondence should be addressed.
Received: 7 April 2013 / Revised: 15 May 2013 / Accepted: 4 June 2013 / Published: 9 July 2013
(This article belongs to the Special Issue Wavelets and Applications)
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Abstract

We study the asymptotics in L2 for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains—e.g., images—by piecewise smooth functions. We introduce a fairly general setting, which comprises most of the presently popular partitions of signal or image domains, like interval, wedgelet or related partitions, as well as Delaunay triangulations. Then, we prove consistency and derive convergence rates. Finally, we illustrate by way of relevant examples that the abstract results are useful for many applications. View Full-Text
Keywords: adaptive estimation; penalized M-estimation; Potts functional; complexity penalized; variational approach; consistency; convergence rates; wedgelet partitions; Delaunay triangulations adaptive estimation; penalized M-estimation; Potts functional; complexity penalized; variational approach; consistency; convergence rates; wedgelet partitions; Delaunay triangulations
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Demaret, L.; Friedrich, F.; Liebscher, V.; Winkler, G. Complexity L0-Penalized M-Estimation: Consistency in More Dimensions. Axioms 2013, 2, 311-344.

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