Open Access
December 2019 Additive models with trend filtering
Veeranjaneyulu Sadhanala, Ryan J. Tibshirani
Ann. Statist. 47(6): 3032-3068 (December 2019). DOI: 10.1214/19-AOS1833


We study additive models built with trend filtering, that is, additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k\geq0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) total variation regularizer that it uses. On the theory side, we derive fast error rates for additive trend filtering estimates, and show these rates are minimax optimal when the underlying function is additive and has component functions whose derivatives are of bounded variation. We also show that these rates are unattainable by additive smoothing splines (and by additive models built from linear smoothers, in general). On the computational side, we use backfitting, to leverage fast univariate trend filtering solvers; we also describe a new backfitting algorithm whose iterations can be run in parallel, which (as far as we can tell) is the first of its kind. Lastly, we present a number of experiments to examine the empirical performance of trend filtering.


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Veeranjaneyulu Sadhanala. Ryan J. Tibshirani. "Additive models with trend filtering." Ann. Statist. 47 (6) 3032 - 3068, December 2019.


Received: 1 April 2018; Revised: 1 February 2019; Published: December 2019
First available in Project Euclid: 31 October 2019

Digital Object Identifier: 10.1214/19-AOS1833

Primary: 62G08 , 62G20

Keywords: Additive models , Minimax rates , Nonparametric regression , parallel backfitting , smoothing splines , Trend filtering

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 6 • December 2019
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