PREDICTION BOUNDS FOR HIGHER ORDER TOTAL VARIATION REGULARIZED LEAST SQUARES

• 成果类型: Article
• 署名作者: Ortelli, Francesco; van de Geer, Sara
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2054
• 发表日期: 2021
• 页码: 2755-2773
• 关键词: selection; RECOVERY

摘要

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 l(1)-penalized least squares estimator with interpolating vectors to upper bound the effective sparsity. This allows one to show that the l(1)-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 is an element of {1, 2, 3, 4} and indicate how it could be derived for general k is an element of N.