# New Risk Bounds for 2D Total Variation Denoising

## TL;DR

This paper investigates the adaptivity of 2D Total Variation Denoising (TVD) for piecewise constant functions, showing it performs better than worst-case guarantees and proposing a data-driven tuning method.

## Contribution

It provides new theoretical risk bounds for TVD when the true function is piecewise constant and introduces a data-driven tuning procedure with strong guarantees.

## Key findings

- TVD performs better on piecewise constant functions than in worst-case scenarios.
- A fully data-driven tuning method for TVD achieves near-optimal risk guarantees.
- Theoretical analysis confirms adaptivity of TVD to certain structured functions.

## Abstract

2D Total Variation Denoising (TVD) is a widely used technique for image denoising. It is also an important nonparametric regression method for estimating functions with heterogenous smoothness. Recent results have shown the TVD estimator to be nearly minimax rate optimal for the class of functions with bounded variation. In this paper, we complement these worst case guarantees by investigating the adaptivity of the TVD estimator to functions which are piecewise constant on axis aligned rectangles. We rigorously show that, when the truth is piecewise constant, the ideally tuned TVD estimator performs better than in the worst case. We also study the issue of choosing the tuning parameter. In particular, we propose a fully data driven version of the TVD estimator which enjoys similar worst case risk guarantees as the ideally tuned TVD estimator.