Frame-constrained Total Variation Regularization for White Noise Regression
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. -risk () up to a logarithmic factor in any dimension . 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.
Cite
@article{arxiv.1807.02038,
title = {Frame-constrained Total Variation Regularization for White Noise Regression},
author = {Miguel del Álamo and Housen Li and Axel Munk},
journal= {arXiv preprint arXiv:1807.02038},
year = {2026}
}
Comments
27 pages main text, 7 pages appendix, 2 figures. In this updated version we have simplied the proof of the upper bound and extended the convergence to the complete range $q\in[1,\infty)$ of $L^q$ risks, rather than the range $q\in(1,1+2/d]$ that we had in the first version. Further, the rates in the extended range are shown to be minimax optimal