ADAPTIVE RISK BOUNDS IN UNIVARIATE TOTAL VARIATION DENOISING AND TREND FILTERING

• 成果类型: Article
• 署名作者: Guntuboyina, Adityanand; Lieu, Donovan; Chatterjee, Sabyasachi; Sen, Bodhisattva
• 署名单位: University of California System; University of California Berkeley; University of Illinois System; University of Illinois Urbana-Champaign; Columbia University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1799
• 发表日期: 2020
• 页码: 205-229
• 关键词: spatial adaptation; regression; splines; estimators; selection

摘要

We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given integer r >= 1, the rth order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute rth order discrete derivatives of the fitted function at the design points. For r = 1, the estimator reduces to total variation regularization which has received much attention in the statistics and image processing literature. In this paper, we study the performance of the trend filtering estimator for every r >= 1, both in the constrained and penalized forms. Our main results show that in the strong sparsity setting when the underlying function is a (discrete) spline with few knots, the risk (under the global squared error loss) of the trend filtering estimator (with an appropriate choice of the tuning parameter) achieves the parametric n(-1)-rate, up to a logarithmic (multiplicative) factor. Our results therefore provide support for the use of trend filtering, for every r >= 1, in the strong sparsity setting.