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February 2014 Adaptive piecewise polynomial estimation via trend filtering
Ryan J. Tibshirani
Ann. Statist. 42(1): 285-323 (February 2014). DOI: 10.1214/13-AOS1189

Abstract

We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339–360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute $k$th order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of $k$th degree spline functions, with adaptively chosen knot points (we say “appear” here as trend filtering estimates are not really functions over continuous domains, and are only defined over the discrete set of inputs). This brings to mind comparisons to other nonparametric regression tools that also produce adaptive splines; in particular, we compare trend filtering to smoothing splines, which penalize the sum of squared derivatives across input points, and to locally adaptive regression splines [Ann. Statist. 25 (1997) 387–413], which penalize the total variation of the $k$th derivative. Empirically, we discover that trend filtering estimates adapt to the local level of smoothness much better than smoothing splines, and further, they exhibit a remarkable similarity to locally adaptive regression splines. We also provide theoretical support for these empirical findings; most notably, we prove that (with the right choice of tuning parameter) the trend filtering estimate converges to the true underlying function at the minimax rate for functions whose $k$th derivative is of bounded variation. This is done via an asymptotic pairing of trend filtering and locally adaptive regression splines, which have already been shown to converge at the minimax rate [Ann. Statist. 25 (1997) 387–413]. At the core of this argument is a new result tying together the fitted values of two lasso problems that share the same outcome vector, but have different predictor matrices.

Citation

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Ryan J. Tibshirani. "Adaptive piecewise polynomial estimation via trend filtering." Ann. Statist. 42 (1) 285 - 323, February 2014. https://doi.org/10.1214/13-AOS1189

Information

Published: February 2014
First available in Project Euclid: 19 March 2014

zbMATH: 1307.62118
MathSciNet: MR3189487
Digital Object Identifier: 10.1214/13-AOS1189

Subjects:
Primary: 62G08, 62G20

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 1 • February 2014
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