G-Norm Properties of Bounded Variation Regularization

Abstract

Recently Y. Meyer derived a characterization of the minimizer of the Rudin-Osher-Fatemi functional in a functional analytical framework. In statistics the discrete version of this functional is used to analyze one dimensional data and belongs to the class of nonparametric regression models. In this work we generalize the functional analytical results of Meyer and apply them to a class of regression models, such as quantile, robust, logistic regression, for the analysis of multidimensional data. The characterization of Y. Meyer and our generalization is based on G-norm properties of the data and the minimizer. A geometric point of view of regression minimization is provided.