Title: On the Complexity of the Weighted Fused Lasso,
Abstract: The solution path of the 1D fused lasso for an $n$-dimensional input is
piecewise linear with $\mathcal{O}(n)$ segments (Hoefling et al. 2010 and
Tibshirani et al 2011). However, existing proofs of this bound do not hold for
the weighted fused lasso. At the same time, results for the generalized lasso,
of which the weighted fused lasso is a special case, allow $\Omega(3^n)$
segments (Mairal et al. 2012). In this paper, we prove that the number of
segments in the solution path of the weighted fused lasso is
$\mathcal{O}(n^2)$, and that, for some instances, it is $\Omega(n^2)$. We also
give a new, very simple, proof of the $\mathcal{O}(n)$ bound for the fused
lasso.