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

## TL;DR

This paper develops oracle inequalities for square root analysis estimators, including total variation penalties on graphs, providing theoretical guarantees and extending previous entropy-based results.

## Contribution

It introduces new oracle inequalities for square root analysis estimators and extends the theory to total variation regularization on graphs.

## Key findings

- Oracle inequalities with fast and slow rates derived for analysis estimators.
- Extension of theory to square root analysis estimators.
- Constant-friendly rates for total variation regularized estimators on graphs.

## Abstract

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.