# New Risk Bounds for 2D Total Variation Denoising

arXiv: https://arxiv.org/abs/1902.01215 · HTML: https://mlsanity.com/paper/1902.01215/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2019.

## Built on by (most cited first, 6 of 6)

- [Multivariate Trend Filtering for Lattice Data](https://mlsanity.com/paper/2112.14758/) (2021)
- [Steered Mixture of Experts Regression for Image Denoising with Multi-Model-Inference](https://mlsanity.com/paper/2303.17409/) (2023)
- [Variance estimation in graphs with the fused lasso](https://mlsanity.com/paper/2207.12638/) (2022)
- [A Cross Validation Framework for Signal Denoising with Applications to Trend Filtering, Dyadic CART and Beyond](https://mlsanity.com/paper/2201.02654/) (2022)
- [Spatially Adaptive Online Prediction of Piecewise Regular Functions](https://mlsanity.com/paper/2203.16587/) (2022)
- [Risk Bounds For Distributional Regression](https://mlsanity.com/paper/2505.09075/) (2025)

## Abstract (arXiv)

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.