Frame-constrained total variation regularization for white noise regression
2021 | journal article. A publication with affiliation to the University of Göttingen.
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- Authors
- del Álamo, Miguel; Li, Housen ; Munk, Axel
- 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. ^qehBrisk (\leq q<\infty$) up to a logarithmic factor in any dimension \geq 1$. 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.
- Issue Date
- 2021
- Journal
- The Annals of Statistics
- Project
- RTG 2088: Research Training Group 2088 Discovering structure in complex data: Statistics meets Optimization and Inverse Problems
EXC 2067: Multiscale Bioimaging - Working Group
- RG Li
RG Munk - ISSN
- 0090-5364