On the Complexity of the Weighted Fussed Lasso

The solution path of the 1D fused lasso for an ndimensional input is piecewise linear with O(n) segments [1], [2]. 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 Ω(3) segments [3]. In this paper, we prove that the number of segments in the solution path of the weighted fused lasso is O(n), and for some instances Ω(n). We also give a new, very simple, proof of the O(n) bound for the fused lasso.