# Manifold optimization for non-linear optimal transport problems

arXiv: https://arxiv.org/abs/2103.00902 · HTML: https://mlsanity.com/paper/2103.00902/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 40 references that are in the map.
- First posted on arXiv 1 Mar 2021.

## Builds on (most cited first, 20 of 20)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Wasserstein regularization for sparse multi-task regression](https://mlsanity.com/paper/1805.07833/) (2018)
- [Regularized Optimal Transport is Ground Cost Adversarial](https://mlsanity.com/paper/2002.03967/) (2020)
- [McTorch, a manifold optimization library for deep learning](https://mlsanity.com/paper/1810.01811/) (2018)
- [Geometry-aware Domain Adaptation for Unsupervised Alignment of Word Embeddings](https://mlsanity.com/paper/2004.08243/) (2020)
- [Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry](https://mlsanity.com/paper/1802.02628/) (2018)
- [Heterogeneous Tensor Decomposition for Clustering via Manifold Optimization](https://mlsanity.com/paper/1504.01777/) (2015)

## Built on by (most cited first, 8 of 8)

- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Riemannian Optimization on Relaxed Indicator Matrix Manifold](https://mlsanity.com/paper/2503.20505/) (2025)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Rieoptax: Riemannian Optimization in JAX](https://mlsanity.com/paper/2210.04840/) (2022)

## Abstract (arXiv)

Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to computationally approach general non-linear OT problems within the framework of Riemannian manifold optimization. The basis of this is the manifold of doubly stochastic matrices (and their generalization). Even though the manifold geometry is not new, surprisingly, its usefulness for solving general non-linear OT problems has not been popular. To this end, we specifically discuss optimization-related ingredients that allow modeling the OT problem on smooth Riemannian manifolds by exploiting the geometry of the search space. We also discuss extensions where we reuse the developed optimization ingredients. We make available the Manifold optimization-based Optimal Transport, or MOT, repository with codes useful in solving OT problems in Python and Matlab. The codes are available at urlhttps://github.com/SatyadevNtv/MOT.