# Riemannian Metric Learning via Optimal Transport

arXiv: https://arxiv.org/abs/2205.09244 · HTML: https://mlsanity.com/paper/2205.09244/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 41 references that are in the map.
- First posted on arXiv 18 May 2022.

## Builds on (most cited first, 14 of 14)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018)
- [A Locally Adaptive Normal Distribution](https://mlsanity.com/paper/1606.02518/) (2016)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Measure-valued spline curves: an optimal transport viewpoint](https://mlsanity.com/paper/1801.03186/) (2018)
- [Ground Metric Learning on Graphs](https://mlsanity.com/paper/1911.03117/) (2019)
- [A mean field game inverse problem](https://mlsanity.com/paper/2007.11551/) (2020)

## Built on by (most cited first, 21 of 21)

- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Learning diffusion at lightspeed](https://mlsanity.com/paper/2406.12616/) (2024)
- [Neural Optimal Transport with Lagrangian Costs](https://mlsanity.com/paper/2406.00288/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry](https://mlsanity.com/paper/2505.08087/) (2025)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [A Riemannian Approach to Ground Metric Learning for Optimal Transport](https://mlsanity.com/paper/2409.10085/) (2024)
- [Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows](https://mlsanity.com/paper/2410.01950/) (2024)
- [Fast Marching Energy CNN](https://mlsanity.com/paper/2306.16109/) (2023)
- [Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data](https://mlsanity.com/paper/2601.18728/) (2026)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systems](https://mlsanity.com/paper/2605.25107/) (2026)
- [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024)
- [Connecting the geometry and dynamics of many-body complex systems with message passing neural operators](https://mlsanity.com/paper/2502.15913/) (2025)
- [Global Ground Metric Learning with Applications to scRNA data](https://mlsanity.com/paper/2506.15383/) (2025)
- [Fiber Bundle Networks: A Geometric Machine Learning Paradigm](https://mlsanity.com/paper/2512.01151/) (2025)
- [Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transport](https://mlsanity.com/paper/2603.01771/) (2026)
- [On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Data](https://mlsanity.com/paper/2604.07671/) (2026)
- [Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transport](https://mlsanity.com/paper/2605.04965/) (2026)
- [Task-Induced Riemannian Metrics for Vision Transformer Feature Spaces](https://mlsanity.com/paper/2609.27988/) (2026)

## Abstract (arXiv)

We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field and efficiently optimize our model's objective using a simple alternating scheme. Using this learned metric, we can nonlinearly interpolate between probability measures and compute geodesics on the manifold. We show that metrics learned using our method improve the quality of trajectory inference on scRNA and bird migration data at the cost of little additional cross-sectional data.