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- W3100222695 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 $L^q$-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." @default.
- W3100222695 created "2020-11-23" @default.
- W3100222695 creator A5039736052 @default.
- W3100222695 creator A5054830960 @default.
- W3100222695 creator A5087767807 @default.
- W3100222695 date "2018-05-01" @default.
- W3100222695 modified "2023-09-27" @default.
- W3100222695 title "Variational multiscale nonparametric regression: Smooth functions" @default.
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- W3100222695 doi "https://doi.org/10.1214/17-aihp832" @default.
- W3100222695 hasPublicationYear "2018" @default.
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