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February 2020 Adaptive risk bounds in univariate total variation denoising and trend filtering
Adityanand Guntuboyina, Donovan Lieu, Sabyasachi Chatterjee, Bodhisattva Sen
Ann. Statist. 48(1): 205-229 (February 2020). DOI: 10.1214/18-AOS1799


We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given integer $r\geq1$, the $r$th 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 $r$th 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 $r\geq1$, 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 $r\geq1$, in the strong sparsity setting.


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Adityanand Guntuboyina. Donovan Lieu. Sabyasachi Chatterjee. Bodhisattva Sen. "Adaptive risk bounds in univariate total variation denoising and trend filtering." Ann. Statist. 48 (1) 205 - 229, February 2020.


Received: 1 February 2017; Revised: 1 June 2018; Published: February 2020
First available in Project Euclid: 17 February 2020

zbMATH: 07196536
MathSciNet: MR4065159
Digital Object Identifier: 10.1214/18-AOS1799

Primary: 62G08 , 62J05 , 62J07

Keywords: Adaptive splines , discrete splines , fat shattering , higher order total variation regularization , metric entropy bounds , nonparametric function estimation , risk bounds , subdifferential , tangent cone

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 1 • February 2020
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