# Untangling Lariats: Subgradient Following of Variationally Penalized Objectives

arXiv: https://arxiv.org/abs/2405.04710 · HTML: https://mlsanity.com/paper/2405.04710/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 28 references that are in the map.
- First posted on arXiv 7 May 2024.

## Builds on (most cited first, 6 of 6)

- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Efficient Implementations of the Generalized Lasso Dual Path Algorithm](https://mlsanity.com/paper/1405.3222/) (2014)
- [Modular proximal optimization for multidimensional total-variation regularization](https://mlsanity.com/paper/1411.0589/) (2014)
- [Total variation on a tree](https://mlsanity.com/paper/1502.07770/) (2015)
- [Sparse regression and marginal testing using cluster prototypes](https://mlsanity.com/paper/1503.00334/) (2015)
- [On the Complexity of the Weighted Fused Lasso](https://mlsanity.com/paper/1801.04987/) (2018)

## Abstract (arXiv)

We describe an apparatus for subgradient-following of the optimum of convex problems with variational penalties. In this setting, we receive a sequence yi,ldots,yn and seek a smooth sequence x1,ldots,xn. The smooth sequence needs to attain the minimum Bregman divergence to an input sequence with additive variational penalties in the general form of sumigi(xi+1-xi). We derive known algorithms such as the fused lasso and isotonic regression as special cases of our approach. Our approach also facilitates new variational penalties such as non-smooth barrier functions. We then derive a novel lattice-based procedure for subgradient following of variational penalties characterized through the output of arbitrary convolutional filters. This paradigm yields efficient solvers for high-order filtering problems of temporal sequences in which sparse discrete derivatives such as acceleration and jerk are desirable. We also introduce and analyze new multivariate problems in which xi,yi∈ℝd with variational penalties that depend on |xi+1-xi|. The norms we consider are ℓ2 and ℓ_∞ which promote group sparsity.