Geometry Lecture (10h) Wasserstein geometry and optimal transport

Max Fathi

• Lecture 1 (2 hours) Introduction to the optimal transport problem on Euclidean space. Formulations of Monge and Kantorovitch, history, applications. Explicit solution in dimension one. Existence of solutions to the Kantorovitch problem.
• Lecture 2(2 hours)  Kantorovitch duality, existence of a transport map solving the Monge problem. Connection with the Monge-Ampere PDE. Extension to Riemannian manifolds.
• Lecture 3 (2 hours) Transport cost as a distance on the space of probability measures, and applications in statistics.
• Lecture 4 (2 hours)  The geometry of optimal transport: Benamou-Brenier formula and Riemannian structure of the space of probability measures. Application: gradient flow structure of the heat equation.
• Lecture 5 (2 hours)  Long-time behavior of stochastic processes, and applications to numerical schemes.

Statistics Lecture (10h) Using the geometry of the Wasserstein space  in statistical learning

Alice Le Brigant

• Lecture 1 (2  hours)  The Fisher information in statistical inference;  Kullback-Leibler divergence; Search for the best estimator: Cramér-Rao  bound and efficient estimators.
  
• Lecture 2 (2 hours) The Fisher information metric; Fisher geometry of Gaussian distributions; Fréchet mean and the Karcher flow algorithm.
  
• Lecture 3 (2 hours) Information geometry manifold; Dual affine connections; Information alpha manifolds; Divergences; Geometric EM algorithm.
  
• Lecture 4 (2 hours) PCA for a closed convex subset of a Hilbert space; The Wasserstein space in dimension 1, the exponential and  logarithm maps, PCA for a closed convex subset of  an Hilbert Spaces, functional PCA in the Wasserstein Space.
  
• Lecture 5 (2 hours) Wasserstein Barycentre; Existence and uniqueness  of empirical and population Wasserstein barycentre; Minimax convergence  rate for estimating the Wasserstein barycentre of random measures on the real line.