# A contribution to Optimal Transport on incomparable spaces

arXiv: https://arxiv.org/abs/2011.04447 · HTML: https://mlsanity.com/paper/2011.04447/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2020.

## Built on by (most cited first, 15 of 15)

- [Cross-Domain Imitation Learning via Optimal Transport](https://mlsanity.com/paper/2110.03684/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Entropic Gromov-Wasserstein between Gaussian Distributions](https://mlsanity.com/paper/2108.10961/) (2021)
- [Structured Transforms Across Spaces with Cost-Regularized Optimal Transport](https://mlsanity.com/paper/2311.05788/) (2023)
- [Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problem](https://mlsanity.com/paper/2303.05978/) (2023)
- [On Probabilistic Embeddings in Optimal Dimension Reduction](https://mlsanity.com/paper/2408.02433/) (2024)
- [Subspace Detours Meet Gromov-Wasserstein](https://mlsanity.com/paper/2110.10932/) (2021)
- [Detecting Backdoor Poisoning Attacks on Deep Neural Networks by Heatmap Clustering](https://mlsanity.com/paper/2204.12848/) (2022)
- [Linear Partial Gromov-Wasserstein Embedding](https://mlsanity.com/paper/2410.16669/) (2024)
- [Model alignment using inter-modal bridges](https://mlsanity.com/paper/2505.12322/) (2025)
- [Sliced Inner Product Gromov-Wasserstein Distances](https://mlsanity.com/paper/2605.08546/) (2026)
- [Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality](https://mlsanity.com/paper/2608.27774/) (2026)
- [Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces](https://mlsanity.com/paper/2407.14957/) (2024)
- [Structured Matching via Cost-Regularized Unbalanced Optimal Transport](https://mlsanity.com/paper/2511.19075/) (2025)
- [Group Contrastive Learning for Weakly Paired Multimodal Data](https://mlsanity.com/paper/2602.04021/) (2026)

## Abstract (arXiv)

Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from this theory, at the frontier between mathematics and optimization. This thesis proposes to study the complex scenario in which the different data belong to incomparable spaces. In particular we address the following questions: how to define and apply Optimal Transport between graphs, between structured data? How can it be adapted when the data are varied and not embedded in the same metric space? This thesis proposes a set of Optimal Transport tools for these different cases. An important part is notably devoted to the study of the Gromov-Wasserstein distance whose properties allow to define interesting transport problems on incomparable spaces. More broadly, we analyze the mathematical properties of the various proposed tools, we establish algorithmic solutions to compute them and we study their applicability in numerous machine learning scenarii which cover, in particular, classification, simplification, partitioning of structured data, as well as heterogeneous domain adaptation.