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