# L1 Regularization Paths in Linear Models by Parametric Gaussian Message Passing

arXiv: https://arxiv.org/abs/2604.16949 · HTML: https://mlsanity.com/paper/2604.16949/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 20 references that are in the map.
- First posted on arXiv 18 Apr 2026.

## Builds on (most cited first, 2 of 2)

- [Efficient Implementations of the Generalized Lasso Dual Path Algorithm](https://mlsanity.com/paper/1405.3222/) (2014)
- [Dual NUP Representations and Min-Maximization in Factor Graphs](https://mlsanity.com/paper/2501.12113/) (2025)

## Abstract (arXiv)

The paper proposes new algorithms for computing the regularization path of linear-model fitting (i) with L1 costs for the independent variables and L2 costs for the dependent variables, and (ii) with L2 costs for the independent variables and L1 costs for the dependent variables. Unlike previously proposed algorithms, the proposed algorithms for (i) and (ii) are exact duals of each other. The heart of the proposed algorithms is parametric Gaussian message passing (i.e., Kalman-type forward-backward recursions) in the pertinent factor graphs. We first develop such algorithms for linear models with general (but preferably low-dimensional) state space representations. We then specialize to specific state space representations of generic linear models; in this case, the structure of the factor graph allows to refine the proposed algorithms such that the computational complexity drops to linear scale.