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

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

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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.
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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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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