Variational multiscale nonparametric regression: Smooth functions.

Grasmair, M.; Li, H.; Munk, A.

doi:10.1214/17-AIHP832

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Variational multiscale nonparametric regression: Smooth functions.

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Li, H.
Research Group of Statistical Inverse-Problems in Biophysics, MPI for Biophysical Chemistry, Max Planck Society;

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Munk, A.
Research Group of Statistical Inverse-Problems in Biophysics, MPI for Biophysical Chemistry, Max Planck Society;

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Grasmair, M., Li, H., & Munk, A. (2018). Variational multiscale nonparametric regression: Smooth functions. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 54(2), 1058-1097. doi: 10.1214/17-AIHP832.

Cite as: https://hdl.handle.net/21.11116/0000-0001-6E8D-6

Abstract

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 (Ann. Statist. 35 (2009) 2313-2351) based on early ideas of Nemirovski (J. Comput. System Sci. 23 (1986) 111). 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 <= q <= infinity, 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. A MATLAB package is available online.