Variational multiscale nonparametric regression: Smooth functions

被引:7
作者
Grasmair, Markus [1 ]
Li, Housen [2 ,3 ]
Munk, Axel [2 ,3 ]
机构
[1] Norwegian Univ Sci & Technol, Dept Math Sci, N-7491 Trondheim, Norway
[2] Univ Gottingen, Inst Math Stochast, Goldschmidtstr 7, D-37077 Gottingen, Germany
[3] Max Planck Inst Biophys Chem, Goldschmidtstr 7, D-37077 Gottingen, Germany
来源
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES | 2018年 / 54卷 / 02期
关键词
Nonparametric regression; Adaptation; Convergence rates; Minimax optimality; Multiresolution norm; Approximate source conditions; LINEAR INVERSE PROBLEMS; L-INFINITY-NORM; CONVERGENCE-RATES; TIKHONOV REGULARIZATION; VARIANCE-ESTIMATION; SPATIAL ADAPTATION; STABLE RECOVERY; WAVELET; SHRINKAGE; PROOF;
D O I
10.1214/17-AIHP832
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
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.
引用
收藏
页码:1058 / 1097
页数:40
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