# Wasserstein regularization for sparse multi-task regression

arXiv: https://arxiv.org/abs/1805.07833 · HTML: https://mlsanity.com/paper/1805.07833/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2018.

## Built on by (most cited first, 18 of 18)

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- [Linear Optimal Partial Transport Embedding](https://mlsanity.com/paper/2302.03232/) (2023)
- [A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions](https://mlsanity.com/paper/2405.03664/) (2024)
- [Sliced Multi-Marginal Optimal Transport](https://mlsanity.com/paper/2102.07115/) (2021)
- [PhysioMTL: Personalizing Physiological Patterns using Optimal Transport Multi-Task Regression](https://mlsanity.com/paper/2203.12595/) (2022)
- [OTCXR: Rethinking Self-supervised Alignment using Optimal Transport for Chest X-ray Analysis](https://mlsanity.com/paper/2404.11868/) (2024)
- [Averaging Spatio-temporal Signals using Optimal Transport and Soft Alignments](https://mlsanity.com/paper/2203.05813/) (2022)
- [What is the Right Notion of Distance between Predict-then-Optimize Tasks?](https://mlsanity.com/paper/2409.06997/) (2024)
- [Towards Dynamic Feature Selection with Attention to Assist Banking Customers in Establishing a New Business](https://mlsanity.com/paper/2105.03852/) (2021)
- [Estimation of Stationary Optimal Transport Plans](https://mlsanity.com/paper/2107.11858/) (2021)
- [Safe Screening for Unbalanced Optimal Transport](https://mlsanity.com/paper/2307.00247/) (2023)
- [Neuroplasticity-inspired dynamic ANNs for multi-task demand forecasting](https://mlsanity.com/paper/2509.24495/) (2025)
- [Optimal Transport Metric Learning for Feature Alignment in Partially Supervised Segmentation](https://mlsanity.com/paper/2609.19176/) (2026)

## Abstract (arXiv)

We focus in this paper on high-dimensional regression problems where each regressor can be associated to a location in a physical space, or more generally a generic geometric space. Such problems often employ sparse priors, which promote models using a small subset of regressors. To increase statistical power, the so-called multi-task techniques were proposed, which consist in the simultaneous estimation of several related models. Combined with sparsity assumptions, it lead to models enforcing the active regressors to be shared across models, thanks to, for instance L1 / Lq norms. We argue in this paper that these techniques fail to leverage the spatial information associated to regressors. Indeed, while sparse priors enforce that only a small subset of variables is used, the assumption that these regressors overlap across all tasks is overly simplistic given the spatial variability observed in real data. In this paper, we propose a convex regularizer for multi-task regression that encodes a more flexible geometry. Our regularizer is based on unbalanced optimal transport (OT) theory, and can take into account a prior geometric knowledge on the regressor variables, without necessarily requiring overlapping supports. We derive an efficient algorithm based on a regularized formulation of OT, which iterates through applications of Sinkhorn's algorithm along with coordinate descent iterations. The performance of our model is demonstrated on regular grids with both synthetic and real datasets as well as complex triangulated geometries of the cortex with an application in neuroimaging.