Oracle inequalities for image denoising with total variation regularization

We derive oracle results for discrete image denoising with a total variation penalty. We consider the least squares estimator with a penalty on the $\ell^1$-norm of the total discrete derivative of the image. This estimator falls into the class of analysis estimators. A bound on the effective sparsity by means of an interpolating matrix allows us to obtain oracle inequalities with fast rates. The bound is an extension of the bound by Ortelli and van de Geer [2019c] to the two-dimensional case. We also present an oracle inequality with slow rates, which matches, up to a log-term, the rate obtained for the same estimator by Mammen and van de Geer [1997]. The key ingredient for our results are the projection arguments to bound the empirical process due to Dalalyan et al. [2017].