# Unbalanced Multi-Marginal Optimal Transport

arXiv: https://arxiv.org/abs/2103.10854 · HTML: https://mlsanity.com/paper/2103.10854/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2021.

## Built on by (most cited first, 11 of 11)

- [The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classification](https://mlsanity.com/paper/2204.12676/) (2022)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Contrasting Multiple Representations with the Multi-Marginal Matching Gap](https://mlsanity.com/paper/2405.19532/) (2024)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization](https://mlsanity.com/paper/2502.07510/) (2025)
- [Fast computation of the TGOSPA metric for multiple target tracking via unbalanced optimal transport](https://mlsanity.com/paper/2503.09449/) (2025)
- [Approximative Algorithms for Multi-Marginal Optimal Transport and Free-Support Wasserstein Barycenters](https://mlsanity.com/paper/2202.00954/) (2022)
- [HOT-POT: Optimal Transport for Sparse Stereo Matching](https://mlsanity.com/paper/2601.12423/) (2026)
- [Locality-aware Private Class Identification for Domain Adaptation with Extreme Label Shift](https://mlsanity.com/paper/2605.05567/) (2026)
- [Path-independent Flow Matching for Multi-parameter Generative Dynamics](https://mlsanity.com/paper/2605.13487/) (2026)

## Abstract (arXiv)

Entropy regularized optimal transport and its multi-marginal generalization have attracted increasing attention in various applications, in particular due to efficient Sinkhorn-like algorithms for computing optimal transport plans. However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual, and show that a unique optimal transport plan exists under mild assumptions. Further, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. If the cost function decouples according to a tree, the iterates can be computed efficiently. At the end, we discuss three applications of our framework, namely two barycenter problems and a transfer operator approach, where we establish a relation between the barycenter problem and the multi-marginal optimal transport with an appropriate tree-structured cost function.