Prediction bounds for higher order total variation regularized least squares
Francesco Ortelli, Sara van de Geer
The Annals of Statistics · 2021 · Volume 49 · Issue 5
Abstract
We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of (k−1)th order differences. Our approach is based on combining a general oracle inequality for the ℓ1-penalized least squares estimator with “interpolating vectors” to upper bound the “effective sparsity.” This allows one to show that the ℓ1-penalty on the kth order differences leads to an estimator that can adapt to the number of jumps in the (k−1)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 k∈N.
Keywords
62J05, 62J99, analysis, compatibility, Lasso, minimax, Moore–Penrose pseudo inverse, Oracle inequality, projection, Total variation regularization
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