A note on the geometry of the multiresolution criterion

Abstract

Several recent developments in nonparametric regression are based on the concept of data approximation: They aim at finding the simplest model that is an adequate approximation to the data. Approximations are regarded as adequate iff the residuals ’look like noise’. This is usually checked with the so-called multiresolution criterion. We show that this criterion is related to a special norm (the ’multiresolution norm’), and point out some important differences between this norm and the p-norms often used to measure the size of residuals. We also treat an important approximation problem with regard to this norm that can be solved using linear programming. Finally, we give sharp upper and lower bounds for the multiresolution norm in terms of p-norms.