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

TL;DR

This paper extends total variation regularization results from path graphs to more complex tree graphs, providing new bounds and insights into the estimator's behavior and conditions for successful recovery.

Contribution

It generalizes oracle inequalities for total variation regularization from path graphs to tree graphs, introducing bounds involving harmonic means and analyzing compatibility constants.

Findings

Derived a lower bound on the compatibility constant for tree graphs.

Extended oracle inequalities to tree graphs with harmonic mean substitution.

Provided insights into the irrepresentable condition for tree-structured graphs.

Abstract

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.