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 n ) segments [3]. In this paper, 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.
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