Oracle inequalities for square root analysis estimators with application to total variation penalties
This work provides theoretical guarantees for estimators in statistical learning, particularly for graph-based regularization, but it appears incremental as it adapts existing arguments to extend theory.
The authors tackled the problem of deriving oracle inequalities for square root analysis estimators, specifically applying them to total variation penalties on graphs, and obtained constant-friendly rates that match previous results up to logarithmic terms.
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.