scholarly journals Adaptive Wavelet Estimations in the Convolution Structure Density Model

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1391
Author(s):  
Kaikai Cao ◽  
Xiaochen Zeng
Keyword(s):  
Kernel Methods ◽  
Convergence Rates ◽  
Adaptive Estimation ◽  
Density Model ◽  
Oracle Inequalities ◽  
Adaptive Wavelet ◽  
Wavelet Estimator ◽  
Convolution Structure ◽  
Optimal Convergence Rates ◽  
Structure Density

Using kernel methods, Lepski and Willer study a convolution structure density model and establish adaptive and optimal Lp risk estimations over an anisotropic Nikol’skii space (Lepski, O.; Willer, T. Oracle inequalities and adaptive estimation in the convolution structure density model. Ann. Stat.2019, 47, 233–287). Motivated by their work, we consider the same problem over Besov balls by wavelets in this paper and first provide a linear wavelet estimate. Subsequently, a non-linear wavelet estimator is introduced for adaptivity, which attains nearly-optimal convergence rates in some cases.

scholarly journals Lower bounds in the convolution structure density model

Bernoulli ◽  
2017 ◽  
Vol 23 (2) ◽  
pp. 884-926 ◽  
Author(s):  
O.V. Lepski ◽  
T. Willer
Keyword(s):  
Lower Bounds ◽  
Density Model ◽  
Convolution Structure ◽  
Structure Density

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

Wavelet regression estimation over Lp risk based on negatively associated sample

2020 ◽  
Vol 18 (04) ◽  
pp. 2050024
Author(s):  
Huijun Guo ◽  
Junke Kou
Keyword(s):  
Convergence Rates ◽  
Regression Function ◽  
Regression Estimation ◽  
Negatively Associated ◽  
Wavelet Regression ◽  
Nonparametric Regression Estimation ◽  
Independent Case ◽  
Optimal Convergence Rates ◽  
Negatively Associated Sample ◽  
Wavelet Estimators

This paper considers wavelet estimations of a regression function based on negatively associated sample. We provide upper bound estimations over [Formula: see text] risk of linear and nonlinear wavelet estimators in Besov space, respectively. When the random sample reduces to the independent case, our convergence rates coincide with the optimal convergence rates of classical nonparametric regression estimation.

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