On the total variation regularized estimator over a class of tree graphs
This work provides theoretical insights for statistical estimation on tree-structured graphs, which is incremental but relevant for applications in network analysis and signal processing.
The authors generalized an oracle inequality for the Fused Lasso from path graphs to tree graphs, showing that the minimum distance between jumps can be replaced by their harmonic mean, and proved a lower bound on the compatibility constant for total variation penalty.
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.