mlsanityData to 7 Oct 2026

Riemannian Metric Learning via Optimal Transport

arXiv 2205.09244 ↗First posted 18 May 2022cs.LG stat.MLcited by 21

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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.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 14 of its 41 references that are in the map. First posted on arXiv 18 May 2022.

Builds on

14 of 41 references.

References · 14

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2017Improved Training of Wasserstein GANscited by 2,223
  3. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  4. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 594
  5. 2020TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamicscited by 155
  6. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 140
  7. 2021Proximal Optimal Transport Modeling of Population Dynamicscited by 91
  8. 2018Scalable Unbalanced Optimal Transport using Generative Adversarial Networkscited by 41
  9. 2016A Locally Adaptive Normal Distributioncited by 23
  10. 2020Fast and Smooth Interpolation on Wasserstein Spacecited by 21
  11. 2018Second order models for optimal transport and cubic splines on the Wasserstein spacecited by 19
  12. 2018Measure-valued spline curves: an optimal transport viewpointcited by 17
2 more
  1. 2019Ground Metric Learning on Graphscited by 17
  2. 2020A mean field game inverse problemcited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 2; 2024: 6; 2025: 6; 2026: 7.

21 papers.

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  1. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  2. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  3. 2024Learning diffusion at lightspeedcited by 21
  4. 2024Neural Optimal Transport with Lagrangian Costscited by 16
  5. 2025Learning of Population Dynamics: Inverse Optimization Meets JKO Schemecited by 6
  6. 2025Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometrycited by 5
  7. 2025Iso-Riemannian Optimization on Learned Data Manifoldscited by 4
  8. 2024A Riemannian Approach to Ground Metric Learning for Optimal Transportcited by 3
  9. 2024Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flowscited by 3
  10. 2023Fast Marching Energy CNNcited by 1
  11. 2026Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Datacited by 1
  12. 2026Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributionscited by 1
9 more
  1. 2026Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systemscited by 1
  2. 2024Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximizationcited by 0
  3. 2025Connecting the geometry and dynamics of many-body complex systems with message passing neural operatorscited by 0
  4. 2025Global Ground Metric Learning with Applications to scRNA datacited by 0
  5. 2025Fiber Bundle Networks: A Geometric Machine Learning Paradigmcited by 0
  6. 2026Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transportcited by 0
  7. 2026On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Datacited by 0
  8. 2026Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transportcited by 0
  9. 2026Task-Induced Riemannian Metrics for Vision Transformer Feature Spacescited by 0