# Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications

arXiv: https://arxiv.org/abs/2311.13883 · HTML: https://mlsanity.com/paper/2311.13883/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 65 of its 621 references that are in the map.
- First posted on arXiv 23 Nov 2023.

## Builds on (most cited first, 30 of 65)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Convergence Rate of Frank-Wolfe for Non-Convex Objectives](https://mlsanity.com/paper/1607.00345/) (2016)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [A Survey on Optimal Transport for Machine Learning: Theory and Applications](https://mlsanity.com/paper/2106.01963/) (2021)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Global convergence of neuron birth-death dynamics](https://mlsanity.com/paper/1902.01843/) (2019)
- [Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning](https://mlsanity.com/paper/1611.01722/) (2016)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [GAN and VAE from an Optimal Transport Point of View](https://mlsanity.com/paper/1706.01807/) (2017)
- [On approximating nabla f with neural networks](https://mlsanity.com/paper/1910.12744/) (2019)
- [Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2303.16510/) (2023)
- [Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation](https://mlsanity.com/paper/1905.10155/) (2019)
- [Input Convex Gradient Networks](https://mlsanity.com/paper/2111.12187/) (2021)
- [On the existence of Monge maps for the Gromov-Wasserstein problem](https://mlsanity.com/paper/2210.11945/) (2022)
- [Sliced Wasserstein Estimation with Control Variates](https://mlsanity.com/paper/2305.00402/) (2023)

## Built on by (most cited first, 1 of 1)

- [Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach](https://mlsanity.com/paper/2602.06539/) (2026)

## Abstract (arXiv)

Optimal Transport has received much attention in Machine Learning as it allows to compare probability distributions by exploiting the geometry of the underlying space. However, in its original formulation, solving this problem suffers from a significant computational burden. Thus, a meaningful line of work consists at proposing alternatives to reduce this burden while still enjoying its properties. In this thesis, we focus on alternatives which use projections on subspaces. The main such alternative is the Sliced-Wasserstein distance, which we first propose to extend to Riemannian manifolds in order to use it in Machine Learning applications for which using such spaces has been shown to be beneficial in the recent years. We also study sliced distances between positive measures in the so-called unbalanced OT problem. Back to the original Euclidean Sliced-Wasserstein distance between probability measures, we study the dynamic of gradient flows when endowing the space with this distance in place of the usual Wasserstein distance. Then, we investigate the use of the Busemann function, a generalization of the inner product in metric spaces, in the space of probability measures. Finally, we extend the subspace detour approach to incomparable spaces using the Gromov-Wasserstein distance.