# Slicing Unbalanced Optimal Transport

arXiv: https://arxiv.org/abs/2306.07176 · HTML: https://mlsanity.com/paper/2306.07176/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2023.

## Built on by (most cited first, 10 of 10)

- [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024)
- [Stereographic Spherical Sliced Wasserstein Distances](https://mlsanity.com/paper/2402.02345/) (2024)
- [Measure transfer via stochastic slicing and matching](https://mlsanity.com/paper/2307.05705/) (2023)
- [PTLp: Partial Transport Lp Distances](https://mlsanity.com/paper/2307.13571/) (2023)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [An Efficient Orlicz-Sobolev Approach for Transporting Unbalanced Measures on a Graph](https://mlsanity.com/paper/2502.00739/) (2025)
- [Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport](https://mlsanity.com/paper/2509.02109/) (2025)
- [Structured Matching via Cost-Regularized Unbalanced Optimal Transport](https://mlsanity.com/paper/2511.19075/) (2025)
- [ASAP: Amortized Doubly-Stochastic Attention via Sliced Dual Projection](https://mlsanity.com/paper/2605.12879/) (2026)
- [Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions](https://mlsanity.com/paper/2606.30310/) (2026)

## Abstract (arXiv)

Optimal transport (OT) is a powerful framework to compare probability measures, a fundamental task in many statistical and machine learning problems. Substantial advances have been made in designing OT variants which are either computationally and statistically more efficient or robust. Among them, sliced OT distances have been extensively used to mitigate optimal transport's cubic algorithmic complexity and curse of dimensionality. In parallel, unbalanced OT was designed to allow comparisons of more general positive measures, while being more robust to outliers. In this paper, we bridge the gap between those two concepts and develop a general framework for efficiently comparing positive measures. We notably formulate two different versions of sliced unbalanced OT, and study the associated topology and statistical properties. We then develop a GPU-friendly Frank-Wolfe like algorithm to compute the corresponding loss functions, and show that the resulting methodology is modular as it encompasses and extends prior related work. We finally conduct an empirical analysis of our loss functions and methodology on both synthetic and real datasets, to illustrate their computational efficiency, relevance and applicability to real-world scenarios including geophysical data.