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Through the direct study of the analysis estimator we derive oracleinequalities with fast and slow rates by adapting the arguments involvingprojections by Dalalyan, Hebiri and Lederer (2017). We then extend the theoryto the square root analysis estimator. Finally, we focus on (square root) totalvariation regularized estimators on graphs and obtain constant-friendly rates,which, up to log-terms, match previous results obtained by entropycalculations. We also obtain an oracle inequality for the (square root) totalvariation regularized estimator over the cycle graph.

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