  Through the direct study of 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 square root analysis estimator. Finally, we focus on (square root) total
variation regularized estimators on graphs and obtain constant-friendly rates,
which, up to log-terms, match previous results obtained by entropy
calculations. We also obtain an oracle inequality for the (square root) total
variation regularized estimator over the cycle graph.
