# CO-Optimal Transport

arXiv: https://arxiv.org/abs/2002.03731 · HTML: https://mlsanity.com/paper/2002.03731/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2020.

## Built on by (most cited first, 30 of 42)

- [Teach a Robot to FISH: Versatile Imitation from One Minute of Demonstrations](https://mlsanity.com/paper/2303.01497/) (2023)
- [Watch and Match: Supercharging Imitation with Regularized Optimal Transport](https://mlsanity.com/paper/2206.15469/) (2022)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Cross-Domain Imitation Learning via Optimal Transport](https://mlsanity.com/paper/2110.03684/) (2021)
- [On Robust Optimal Transport: Computational Complexity and Barycenter Computation](https://mlsanity.com/paper/2102.06857/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Weisfeiler-Lehman meets Gromov-Wasserstein](https://mlsanity.com/paper/2202.02495/) (2022)
- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)
- [The Z-Gromov-Wasserstein Distance](https://mlsanity.com/paper/2408.08233/) (2024)
- [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025)
- [Manifold optimization for non-linear optimal transport problems](https://mlsanity.com/paper/2103.00902/) (2021)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [LiveSeg: Unsupervised Multimodal Temporal Segmentation of Long Livestream Videos](https://mlsanity.com/paper/2210.05840/) (2022)
- [Block-coordinate Frank-Wolfe algorithm and convergence analysis for semi-relaxed optimal transport problem](https://mlsanity.com/paper/2205.13766/) (2022)
- [Unsupervised Manifold Alignment with Joint Multidimensional Scaling](https://mlsanity.com/paper/2207.02968/) (2022)
- [Match-And-Deform: Time Series Domain Adaptation through Optimal Transport and Temporal Alignment](https://mlsanity.com/paper/2308.12686/) (2023)
- [Leveraging Optimal Transport for Enhanced Offline Reinforcement Learning in Surgical Robotic Environments](https://mlsanity.com/paper/2310.08841/) (2023)
- [Fast block-coordinate Frank-Wolfe algorithm for semi-relaxed optimal transport](https://mlsanity.com/paper/2103.05857/) (2021)
- [Quantum Theory and Application of Contextual Optimal Transport](https://mlsanity.com/paper/2402.14991/) (2024)
- [Factored couplings in multi-marginal optimal transport via difference of convex programming](https://mlsanity.com/paper/2110.00629/) (2021)
- [Learning representations that are closed-form Monge mapping optimal with application to domain adaptation](https://mlsanity.com/paper/2305.07500/) (2023)
- [Fast Gradient Computation for Gromov-Wasserstein Distance](https://mlsanity.com/paper/2404.08970/) (2024)
- [Metric properties of partial and robust Gromov-Wasserstein distances](https://mlsanity.com/paper/2411.02198/) (2024)
- [Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization](https://mlsanity.com/paper/2502.07510/) (2025)
- [Wasserstein Distances Made Explainable: Insights Into Dataset Shifts and Transport Phenomena](https://mlsanity.com/paper/2505.06123/) (2025)
- [Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks](https://mlsanity.com/paper/2508.10888/) (2025)
- [Hybrid Gromov-Wasserstein Embedding for Capsule Learning](https://mlsanity.com/paper/2209.00232/) (2022)
- [Unsupervised Domain Adaptation for Extra Features in the Target Domain Using Optimal Transport](https://mlsanity.com/paper/2209.04594/) (2022)
- [Revisiting invariances and introducing priors in Gromov-Wasserstein distances](https://mlsanity.com/paper/2307.10093/) (2023)
- [Graph Cuts with Arbitrary Size Constraints Through Optimal Transport](https://mlsanity.com/paper/2402.04732/) (2024)

## Abstract (arXiv)

Optimal transport (OT) is a powerful geometric and probabilistic tool for finding correspondences and measuring similarity between two distributions. Yet, its original formulation relies on the existence of a cost function between the samples of the two distributions, which makes it impractical when they are supported on different spaces. To circumvent this limitation, we propose a novel OT problem, named COOT for CO-Optimal Transport, that simultaneously optimizes two transport maps between both samples and features, contrary to other approaches that either discard the individual features by focusing on pairwise distances between samples or need to model explicitly the relations between them. We provide a thorough theoretical analysis of our problem, establish its rich connections with other OT-based distances and demonstrate its versatility with two machine learning applications in heterogeneous domain adaptation and co-clustering/data summarization, where COOT leads to performance improvements over the state-of-the-art methods.