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

TL;DR

This paper investigates the asymptotic properties of $L^0$-penalized M-estimation for piecewise smooth signal approximation, establishing consistency and convergence rates across various partitioning methods.

Contribution

It introduces a general framework for complexity penalized least squares regression applicable to multiple partitioning schemes and proves their consistency and convergence rates.

Findings

Proves consistency of $L^0$-penalized estimators.

Derives convergence rates for various partitions.

Demonstrates applicability to practical signal and image processing examples.

Abstract

We study the asymptotics in $L^2$ 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.