Crash Course on Optimal Transport

on YouTube ↗01h 02m

" summary

Covers Monge vs Kantorovich formulations and shows that Monge maps can be empty while Kantorovich couplings always exist, because Kantorovich turns transport into a linear program with convex marginals. States Kantorovich duality and the simple dual for cost |x−y|: a supremum over 1‑Lipschitz potentials. Proves Brenier (called Wren) for quadratic cost: if mu is absolutely continuous then the optimal coupling is unique and equals a map T = ∇φ with φ convex. Defines p‑Wasserstein, geodesics via linear interpolation and constant‑speed property.

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" chapters (10)

1. 0:00Introduction
2. 3:25Brief overview
3. 8:28Cost formulation
4. 12:13Applications
5. 15:45Definitions
6. 20:01Question
7. 31:36Structure
8. 35:37Optimal Coupling
9. 50:04Moment Condition
10. 56:14Optimal Transport Map

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