Adaptive risk bounds in univariate total variation denoising and trend filtering
| Autor: | Donovan Lieu, Bodhisattva Sen, Sabyasachi Chatterjee, Adityanand Guntuboyina |
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| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Mean squared error Logarithm fat shattering Mathematics - Statistics Theory Statistics Theory (math.ST) 01 natural sciences 010104 statistics & probability 62J07 62G08 62J05 FOS: Mathematics Applied mathematics 0101 mathematics tangent cone Mathematics Parametric statistics risk bounds Multiplicative function higher order total variation regularization Univariate subdifferential Estimator Adaptive splines discrete splines Total variation denoising Nonparametric regression metric entropy bounds Statistics Probability and Uncertainty nonparametric function estimation |
| Zdroj: | Ann. Statist. 48, no. 1 (2020), 205-229 |
| ISSN: | 0090-5364 |
| DOI: | 10.1214/18-aos1799 |
| Popis: | We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given positive integer $r$, 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 positive integer $r$, 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 $r$, in the strong sparsity setting. Comment: 110 pages, 8 figures |
| Databáze: | OpenAIRE |
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