Variational multiscale nonparametric regression: Smooth functions
2018 | journal article. A publication with affiliation to the University of Göttingen.
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- Authors
- Grasmair, Markus; Li, Housen ; Munk, Axel
- 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 (\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 Lq-loss, 1≤q≤∞, 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.
- Issue Date
- 2018
- Journal
- Annales de l´Institut Henri Poincaré. B, Probability and Statistics
- Project
- RTG 2088: Research Training Group 2088 Discovering structure in complex data: Statistics meets Optimization and Inverse Problems
- ISSN
- 0246-0203
- Language
- English