Frame-constrained Total Variation Regularization for White Noise Regression

TL;DR

This paper proves that frame-constrained total variation estimators are minimax optimal for white noise regression in any dimension, confirming their practical effectiveness through a new theoretical connection to Besov norms.

Contribution

It establishes the minimax optimality of frame-constrained TV estimators in high dimensions, linking frame constraints to Besov norms and risk analysis.

Findings

Frame-constrained TV estimators are minimax optimal up to a logarithmic factor.

Theoretical results confirm practical success of overcomplete frames with TV regularization.

A novel connection between frame-constraints and Besov norms underpins the analysis.

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. $L^q$-risk ($1\leq q<\infty$) up to a logarithmic factor in any dimension $d\geq 1$. 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.