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.Comment: 42 page
