# On the total variation regularized estimator over a class of tree graphs

arXiv: https://arxiv.org/abs/1806.01009 · HTML: https://mlsanity.com/paper/1806.01009/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2018.

## Built on by (most cited first, 2 of 2)

- [Variance estimation in graphs with the fused lasso](https://mlsanity.com/paper/2207.12638/) (2022)
- [Denoising and change point localisation in piecewise-constant high-dimensional regression coefficients](https://mlsanity.com/paper/2110.14298/) (2021)

## Abstract (arXiv)

We generalize to tree graphs obtained by connecting path graphs an oracle result obtained for the Fused Lasso over the path graph. Moreover we show that it is possible to substitute in the oracle inequality the minimum of the distances between jumps by their harmonic mean. In doing so we prove a lower bound on the compatibility constant for the total variation penalty. Our analysis leverages insights obtained for the path graph with one branch to understand the case of more general tree graphs. As a side result, we get insights into the irrepresentable condition for such tree graphs.