Oracle inequalities for square root analysis estimators with application to total variation penalties

We study the analysis estimator directly, without any step through a synthesis formulation. For the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the case of the square root analysis estimator. Finally, we narrow down our attention to a particular class of analysis estimators: (square root) total variation regularized estimators on graphs. In this case, we obtain constant-friendly rates which match up to log-terms previous results obtained by entropy calculations. Moreover, we obtain an oracle inequality for the (square root) total variation regularized estimator over the cycle graph.