DRP (IAP 2020)

This is the webpage for a directed reading program (DRP) I am leading during MIT's Independent Activities Period (IAP 2020). The subject is high-dimensional statistics and an introduction to optimal transport.

References

  • [PC] Peyré and Cuturi, Computational Optimal Transport (2019).
  • [W] Wainwright, High-Dimensional Statistics (2019).
  • [BBI] Burago, Burago, and Ivanov, A Course in Metric Geometry (2001).
  • [DPV] Dasgupta, Papadimitriou, and Vazirani, Algorithms (2008).
  • [LPW] Levin, Peres, and Wilmer, Markov Chains & Mixing Times (2009).
  • [MM] Moore and Mertens, The Nature of Computation (2011).
  • [S] Santambrogio, Optimal Transport for Applied Mathematicians (2015).
  • [V] Villani, Topics in Optimal Transportation (2003).

Readings

  • (preparation) A preview of high-dimensional statistics. [W] 1.1-1.5.
  • (1/6) Total variation distance and couplings. Monge problem and Kantorovich problem. [LPW] 4.1-4.2, [PC] 2.1-2.3.
  • (1/8) Metric properties of the Wasserstein distance and examples. Introduction to linear programming (LP) duality. [PC] 2.4, 2.6, [DPV] 7.4.
  • (1/10) Kantorovich duality, c-transforms, complementary slackness. [PC] 2.5, 3.1-3.3, 5.1.
  • (1/13) The transportation polytope and its vertices. Simplex algorithm. [PC] 3.4-3.5, [MM] 9.4.1-9.4.3.
  • (1/15) Entropically regularized optimal transport and Sinkhorn's algorithm. [PC] 4.1-4.2.
  • (1/17) W1 optimal transport on metric spaces and its dual. Lipschitz functions and concentration of measure (blow-up). T1 transportation inequality. [PC] 6.1-6.3, [W] 3.2.1-3.2.2, 3.3.1-3.3.2.
  • (1/20) Holiday.
  • (1/22) Continuity equation. Geodesics in Wasserstein space; McCann interpolation and Benamou-Brenier formula. Displacement/geodesic convexity. [PC] 7.1, 7.5, [V] 5.1.3-5.2.1, 8.1.2-8.1.3.
  • (1/24) Survey of ``distances'' between probability distributions: divergences, integral probability metrics, kernels. [PC] 8.1-8.3.
  • (1/27) Wasserstein barycenters and multi-marginal optimal transport. [PC] 9.2, 10.1, [S] 5.5.5.
  • (1/29) Distances between metric spaces: Lipschitz distance, Hausdorff distance, Gromov-Hausdorff distance, Gromov-Wasserstein distance. [PC] 10.6, [BBI] 7.2-7.3.

Exercises