June 2021 Frame-constrained total variation regularization for white noise regression
Miguel del Álamo, Housen Li, Axel Munk
Author Affiliations +
Ann. Statist. 49(3): 1318-1346 (June 2021). DOI: 10.1214/20-AOS2001

Abstract

Despite the popularity and practical success of total variation (TV) regularization for function estimation, surprisingly little is known about its theoretical performance in a statistical setting. While TV regularization has been known for quite some time to be minimax optimal for denoising one-dimensional signals, for higher dimensions this remains elusive until today. In this paper, we consider frame-constrained TV estimators including many well-known (overcomplete) frames in a white noise regression model, and prove their minimax optimality w.r.t. Lq-risk (1q<) up to a logarithmic factor in any dimension d1. Overcomplete frames are an established tool in mathematical imaging and signal recovery, and their combination with TV regularization has been shown to give excellent results in practice, which our theory now confirms. Our results rely on a novel connection between frame-constraints and certain Besov norms, and on an interpolation inequality to relate them to the risk functional. Additionally, our results explain a phase transition in the minimax risk for BV functions.

Funding Statement

The first author was supported by DFG RTG 2088-B2.
The second and third authors were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2067/1-390729940.
The second author was supported by DFG CRC 937-A10.
The third author was supported by DFG CRC 755-A4.

Acknowledgments

We thank two anonymous referees for their insightful comments that improved the scope and quality of the paper.

Citation

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Miguel del Álamo. Housen Li. Axel Munk. "Frame-constrained total variation regularization for white noise regression." Ann. Statist. 49 (3) 1318 - 1346, June 2021. https://doi.org/10.1214/20-AOS2001

Information

Received: 1 May 2019; Revised: 1 January 2020; Published: June 2021
First available in Project Euclid: 9 August 2021

MathSciNet: MR4298866
zbMATH: 1475.62145
Digital Object Identifier: 10.1214/20-AOS2001

Subjects:
Primary: 62G05
Secondary: 62G20 , 62M40

Keywords: interpolation inequalities , minimax estimation , Nonparametric regression , overcomplete dictionaries , Total variation , Wavelets

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.49 • No. 3 • June 2021
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