  The graph fused lasso -- which includes as a special case the one-dimensional
fused lasso -- is widely used to reconstruct signals that are piecewise
constant on a graph, meaning that nodes connected by an edge tend to have
identical values. We consider testing for a difference in the means of two
connected components estimated using the graph fused lasso. A naive procedure
such as a z-test for a difference in means will not control the selective Type
I error, since the hypothesis that we are testing is itself a function of the
data. In this work, we propose a new test for this task that controls the
selective Type I error, and conditions on less information than existing
approaches, leading to substantially higher power. We illustrate our approach
in simulation and on datasets of drug overdose death rates and teenage birth
rates in the contiguous United States. Our approach yields more discoveries on
both datasets.
