Optimal Transport Wiki

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The optimal transportation problem

• Monge Problem
• Monge Problem(revised)
• Kantorovich Problem <— THIS PAGE ONLY
• Optimal Transport in One Dimension
• Kantorovich Dual Problem (for general costs)
• Kantorovich Dual Problem (for c(x,y) = d(x,y)^2 where d is a metric)
• Regularity of Optimal Transport Maps and the Monge-Ampére Equation on Riemannian Manifolds
• 1-Wasserstein metric and generalizations
• Optimal Transport and Ricci curvature

Variants of the optimal transport problem

• Martingale optimal transport and mathematical finance ¹
• Wasserstein barycenters and applications in image processing

The 2-Wasserstein Metric

• Geodesics and generalized geodesics
• Formal Riemannian Structure of the Wasserstein metric
• Asymptotic equivalence of W_2 and H^-1

Numerical methods for optimal transport

• Discrete Optimal Transport
• Auction Algorithm
• Semidiscrete Optimal Transport
• Sinkhorn’s Algorithm
• Sliced Wasserstein Distance

Mathematical foundations

• Dual space of C_0(x) vs C_b(x)
• Convergence of Measures and Metrizability
• Fenchel-Moreau and Primal/Dual Optimization Problems
• Fenchel-Rockafellar and Linear Programming
• The Moreau-Yosida Regularization
• Gradient flows in Hilbert spaces
• The continuity equation and Benamour Brenier formula
• Isoperimetric inequality and OMT

Applications of Optimal Transport

• Machine Learning
• Shallow neural networks as Wasserstein gradient flows

Other

• Distraction Recommendations
• Nonlocal Equations Wiki
• Proteopedia
• Quantum Information Wiki

Footnotes

1. Santambrogio (51-53); https://link.springer.com/content/pdf/10.1007/s00440-013-0531-y.pdf