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Complexity L0-Penalized M-Estimation: Consistency in More Dimensions
Institute of Computational Biology, German Research Center for Environmental Health, Ingolstädter Landstr. 1, Neuherberg D-85764, Germany
Computer Systems Institute, Swiss Federal Institute of Technology (ETH), Zürich 8092, Switzerland
Department of Mathematics and Computer Science, University of Greifswald, Domstr. 11, Greifswald 17489, Germany
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; in revised form: 15 May 2013 / Accepted: 4 June 2013 / Published: 9 July 2013
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
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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.
Demaret L, Friedrich F, Liebscher V, Winkler G. Complexity L0-Penalized M-Estimation: Consistency in More Dimensions. Axioms. 2013; 2(3):311-344.
Demaret, Laurent; Friedrich, Felix; Liebscher, Volkmar; Winkler, Gerhard. 2013. "Complexity L0-Penalized M-Estimation: Consistency in More Dimensions." Axioms 2, no. 3: 311-344.