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.

We generalize to tree graphs obtained by connecting path graphs an oracle result obtained for the Fused Lasso over the path graph. Moreover we show that it is possible to substitute in the oracle inequality the minimum of the distances between jumps by their harmonic mean. In doing so we prove a lower bound on the compatibility constant for the total variation penalty. Our analysis leverages insights obtained for the path graph with one branch to understand the case of more general tree graphs. As a side result, we get insights into the irrepresentable condition for such tree graphs.