Frame-constrained total variation regularization for white noise regression
Miguel del Álamo, Housen Li, Axel Munk
The Annals of Statistics · 2021 · Volume 49 · Issue 3
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 (1≤q<∞) up to a logarithmic factor in any dimension d≥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. Additionally, our results explain a phase transition in the minimax risk for BV functions.
Keywords
62G05, 62G20, 62M40, interpolation inequalities, minimax estimation, Nonparametric regression, overcomplete dictionaries, Total variation, Wavelets
Enable JavaScript for citation tools and additional paper interactions.