This paper formulates a general cross validation framework for signal denoising. The general framework is then applied to nonparametric regression methods such as Trend Filtering and Dyadic CART. The resulting cross validated versions are then shown to attain nearly the same rates of convergence as are known for the optimally tuned analogues. There did not exist any previous theoretical analyses of cross validated versions of Trend Filtering or Dyadic CART. To illustrate the generality of the framework we also propose and study cross validated versions of two fundamental estimators; lasso for high dimensional linear regression and singular value thresholding for matrix estimation. Our general framework is inspired by the ideas in Chatterjee and Jafarov (2015) and is potentially applicable to a wide range of estimation methods which use tuning parameters.

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].