On the Complexity of the Weighted Fused Lasso

The solution path of the one-dimensional fused lasso for an n-dimensional input is piecewise linear with at most O(n) segments. 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 more than Ω(3
ⁿ
) segments. In this letter, we prove that the number of segments in the solution path of the weighted fused lasso is O(n
²
), and that, for some instances, it is Ω(n
²
). We also give a new, very simple, proof of the O(n) bound for the fused lasso.