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We study the asymptotics in $L^2$ for complexity penalized least squaresregression for the discrete approximation of finite-dimensional signals oncontinuous domains - e.g. images - by piecewise smooth functions.  We introduce a fairly general setting which comprises most of the presentlypopular partitions of signal- or image- domains like interval-, wedgelet- orrelated partitions, as well as Delaunay triangulations. Then we proveconsistency and derive convergence rates. Finally, we illustrate by way ofrelevant examples that the abstract results are useful for many applications.

Complexity $L^0$-penalized M-Estimation: Consistency in More Dimensions