# Towards Optimal Transport with Global Invariances

arXiv: https://arxiv.org/abs/1806.09277 · HTML: https://mlsanity.com/paper/1806.09277/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2018.

## Built on by (most cited first, 30 of 33)

- [Relative representations enable zero-shot latent space communication](https://mlsanity.com/paper/2209.15430/) (2022)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Cross-Domain Imitation Learning via Optimal Transport](https://mlsanity.com/paper/2110.03684/) (2021)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Pre-training for Speech Translation: CTC Meets Optimal Transport](https://mlsanity.com/paper/2301.11716/) (2023)
- [Using Optimal Transport as Alignment Objective for fine-tuning Multilingual Contextualized Embeddings](https://mlsanity.com/paper/2110.02887/) (2021)
- [Soft Matching Distance: A metric on neural representations that captures single-neuron tuning](https://mlsanity.com/paper/2311.09466/) (2023)
- [InfoOT: Information Maximizing Optimal Transport](https://mlsanity.com/paper/2210.03164/) (2022)
- [E(n)-equivariant Graph Neural Cellular Automata](https://mlsanity.com/paper/2301.10497/) (2023)
- [Structured Transforms Across Spaces with Cost-Regularized Optimal Transport](https://mlsanity.com/paper/2311.05788/) (2023)
- [Functional optimal transport: map estimation and domain adaptation for functional data](https://mlsanity.com/paper/2102.03895/) (2021)
- [Unsupervised Manifold Alignment with Joint Multidimensional Scaling](https://mlsanity.com/paper/2207.02968/) (2022)
- [Personalised Federated Learning On Heterogeneous Feature Spaces](https://mlsanity.com/paper/2301.11447/) (2023)
- [Latent Space Alignment for AI-Native MIMO Semantic Communications](https://mlsanity.com/paper/2507.16680/) (2025)
- [Order Constraints in Optimal Transport](https://mlsanity.com/paper/2110.07275/) (2021)
- [Subspace Detours Meet Gromov-Wasserstein](https://mlsanity.com/paper/2110.10932/) (2021)
- [Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space](https://mlsanity.com/paper/2310.11256/) (2023)
- [Explanatory Model Monitoring to Understand the Effects of Feature Shifts on Performance](https://mlsanity.com/paper/2408.13648/) (2024)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Learning Network Sheaves for AI-native Semantic Communication](https://mlsanity.com/paper/2512.03248/) (2025)
- [On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport](https://mlsanity.com/paper/2109.15030/) (2021)
- [PhysioMTL: Personalizing Physiological Patterns using Optimal Transport Multi-Task Regression](https://mlsanity.com/paper/2203.12595/) (2022)
- [Relative Translation Invariant Wasserstein Distance](https://mlsanity.com/paper/2409.02416/) (2024)
- [Graph Matching Optimization Network for Point Cloud Registration](https://mlsanity.com/paper/2302.05844/) (2023)
- [Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization](https://mlsanity.com/paper/2502.07510/) (2025)
- [An in depth look at the Procrustes-Wasserstein distance: properties and barycenters](https://mlsanity.com/paper/2507.00894/) (2025)
- [Optimal transport for automatic alignment of untargeted metabolomic data](https://mlsanity.com/paper/2306.03218/) (2023)
- [Covariance alignment: from maximum likelihood estimation to Gromov-Wasserstein](https://mlsanity.com/paper/2311.13595/) (2023)
- [Federated Latent Space Alignment for Multi-user Semantic Communications](https://mlsanity.com/paper/2602.17271/) (2026)

## Abstract (arXiv)

Many problems in machine learning involve calculating correspondences between sets of objects, such as point clouds or images. Discrete optimal transport provides a natural and successful approach to such tasks whenever the two sets of objects can be represented in the same space, or at least distances between them can be directly evaluated. Unfortunately neither requirement is likely to hold when object representations are learned from data. Indeed, automatically derived representations such as word embeddings are typically fixed only up to some global transformations, for example, reflection or rotation. As a result, pairwise distances across two such instances are ill-defined without specifying their relative transformation. In this work, we propose a general framework for optimal transport in the presence of latent global transformations. We cast the problem as a joint optimization over transport couplings and transformations chosen from a flexible class of invariances, propose algorithms to solve it, and show promising results in various tasks, including a popular unsupervised word translation benchmark.