mlsanityData to 7 Oct 2026

Natural gradient via optimal transport

arXiv 1803.07033 ↗First posted 16 Mar 2018math.OC cs.IT math.ITcited by 19

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Abstract (arXiv)

We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient flow there. In contrast to the natural Fisher-Rao gradient, the natural Wasserstein gradient incorporates a ground metric on sample space. We illustrate the analysis of elementary exponential family examples and demonstrate an application of the Wasserstein natural gradient to maximum likelihood estimation.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Mar 2018.

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Citing papers per year: 2021: 4; 2022: 4; 2023: 1; 2024: 5; 2025: 2; 2026: 3.

19 papers.

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  1. 2021Sinkhorn Distributionally Robust Optimizationcited by 38
  2. 2021Wasserstein Proximal of GANscited by 15
  3. 2022Geometry and convergence of natural policy gradient methodscited by 12
  4. 2022Invariance Properties of the Natural Gradient in Overparametrised Systemscited by 11
  5. 2023Achieving High Accuracy with PINNs via Energy Natural Gradientscited by 10
  6. 2022Efficient Natural Gradient Descent Methods for Large-Scale PDE-Based Optimization Problemscited by 7
  7. 2024Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Datacited by 6
  8. 2021Depth Without the Magic: Inductive Bias of Natural Gradient Descentcited by 4
  9. 2021Transport information Bregman divergencescited by 3
  10. 2022Generalized Tangent Kernel: A Unified Geometric Foundation for Natural Gradient and Standard Gradientcited by 2
  11. 2024Gradient flow in parameter space is equivalent to linear interpolation in output spacecited by 2
  12. 2024Sparse Variational Student-t Processes for Heavy-tailed Modelingcited by 2
7 more
  1. 2024Newton Losses: Using Curvature Information for Learning with Differentiable Algorithmscited by 2
  2. 2024A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equationscited by 2
  3. 2025Manifold learning in metric spacescited by 2
  4. 2025ENIGMA: The Geometry of Reasoning and Alignment in Large-Language Modelscited by 1
  5. 2026A Transport-Based Geometry of Belief-Costcited by 0
  6. 2026Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solverscited by 0
  7. 2026The Dually Flat Geometry of Planning as Inferencecited by 0