# On the Complexity of the Weighted Fused Lasso

arXiv: https://arxiv.org/abs/1801.04987 · HTML: https://mlsanity.com/paper/1801.04987/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2018.

## Built on by (most cited first, 2 of 2)

- [Robust Regression over Averaged Uncertainty](https://mlsanity.com/paper/2311.06960/) (2023)
- [Untangling Lariats: Subgradient Following of Variationally Penalized Objectives](https://mlsanity.com/paper/2405.04710/) (2024)

## Abstract (arXiv)

The solution path of the 1D fused lasso for an n-dimensional input is piecewise linear with 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(3n) 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 O(n2), and that, for some instances, it is Omega(n2). We also give a new, very simple, proof of the O(n) bound for the fused lasso.