Regularization of Wavelet Approximations
小波逼近的正则化AI 解读 ↓
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小波逼近的正则化
- 研究问题
- 论文研究采样位置不均匀时如何估计非参数回归函数，并关注惩罚形式如何影响阈值性质、统计表现与计算可行性。
- 方法与思路
- 作者提出非线性正则化小波估计，以惩罚最小二乘为基础，结合正则化插值初值、一步更新及渐进非凸算法求解。
- 材料陈述的贡献
- 摘要称获得多类惩罚的阈值条件、预言机不等式及通用阈值参数，并建立估计量的自适应极小极大性质。
- 阅读关注点
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原文摘要
In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone are specific members of nonlinear regularized wavelet estimators. They correspond to the lower and upper envelopes of a class of the penalized least squares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax. To efficiently solve penalized least squares problems, nonlinear regularized Sobolev interpolators (NRSI) are proposed as initial estimators, which are shown to have good sampling properties. The NRSI is further ameliorated by regularized one-step estimators, which are the one-step estimators of the penalized least squares problems using the NRSI as initial estimators. The graduated nonconvexity algorithm is also introduced to handle penalized least squares problems. The newly introduced approaches are illustrated by a few numerical examples.