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October 2021 Prediction bounds for higher order total variation regularized least squares
Francesco Ortelli, Sara van de Geer
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Ann. Statist. 49(5): 2755-2773 (October 2021). DOI: 10.1214/21-AOS2054

Abstract

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of (k1)th order differences. Our approach is based on combining a general oracle inequality for the 1-penalized least squares estimator with “interpolating vectors” to upper bound the “effective sparsity.” This allows one to show that the 1-penalty on the kth order differences leads to an estimator that can adapt to the number of jumps in the (k1)th order differences of the underlying signal or an approximation thereof. We show the result for k{1,2,3,4} and indicate how it could be derived for general kN.

Funding Statement

We acknowledge support for this project from the the Swiss National Science Foundation (SNF Grant 200020_169011).

Acknowledgments

We thank the Associate Editor and the referees for their very helpful remarks.

Citation

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Francesco Ortelli. Sara van de Geer. "Prediction bounds for higher order total variation regularized least squares." Ann. Statist. 49 (5) 2755 - 2773, October 2021. https://doi.org/10.1214/21-AOS2054

Information

Received: 1 July 2020; Revised: 1 January 2021; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/21-AOS2054

Subjects:
Primary: 62J05
Secondary: 62J99

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • October 2021
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