Variational Multiscale Nonparametric Regression: Smooth Functions

For the problem of nonparametric regression of smooth functions, we reconsider and analyze a constrained variational approach, which we call the MultIscale Nemirovski-Dantzig (MIND) estimator. This can be viewed as a multiscale extension of the Dantzig selector (\emph{Ann. Statist.}, 35(6): 2313--51, 2009) based on early ideas of Nemirovski (\emph{J. Comput. System Sci.}, 23:1--11, 1986). MIND minimizes a homogeneous Sobolev norm under the constraint that the multiresolution norm of the residual is bounded by a universal threshold. The main contribution of this paper is the derivation of convergence rates of MIND with respect to $L^q$-loss, $1 \le q \le \infty$, both almost surely and in expectation. To this end, we introduce the method of approximate source conditions. For a one-dimensional signal, these can be translated into approximation properties of $B$-splines. A remarkable consequence is that MIND attains almost minimax optimal rates simultaneously for a large range of Sobolev and Besov classes, which provides certain adaptation. Complimentary to the asymptotic analysis, we examine the finite sample performance of MIND by numerical simulations.