Optimal rates for total variation denoising

Abstract

Motivated by its practical success, we show that the 2D total variation denoiser satisfies a sharp oracle inequality that leads to near optimal rates of estimation for a large class of image models such as bi-isotonic, Hölder smooth and cartoons. Our analysis hinges on properties of the unnormalized Laplacian of the two-dimensional grid such as eigenvector delocalization and spectral decay. We also present extensions to more than two dimensions as well as several other graphs. Key words and phrases: Total variation regularization; TV denoising; sharp oracle inequalities; image denoising; edge Lasso; trend filtering; nonparametric regression; shape constrained regression; minimax