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Statistical optimal transport

Sinho Chewi, Jonathan Niles-Weed, Philippe Rigollet

TL;DR

This work surveys statistical optimal transport by tracing the historical Monge and Kantorovich formulations and the central role of couplings, culminating in the Wasserstein p-distance framework. It connects OT to geometry via Brenier's theorem and duality, and it sets the stage for statistical applications by highlighting how Wasserstein distances, dual potentials, and transport maps enable principled comparisons of probability measures. The text further motivates computational OT and discusses foundational results that underpin statistical estimation, concentration, and algorithmic approaches in high dimensions. Overall, it builds a bridge from classical OT theory to modern statistical and machine-learning methodologies grounded in optimal transport geometry.

Abstract

We present an introduction to the field of statistical optimal transport, based on lectures given at École d'Été de Probabilités de Saint-Flour XLIX.

Statistical optimal transport

TL;DR

This work surveys statistical optimal transport by tracing the historical Monge and Kantorovich formulations and the central role of couplings, culminating in the Wasserstein p-distance framework. It connects OT to geometry via Brenier's theorem and duality, and it sets the stage for statistical applications by highlighting how Wasserstein distances, dual potentials, and transport maps enable principled comparisons of probability measures. The text further motivates computational OT and discusses foundational results that underpin statistical estimation, concentration, and algorithmic approaches in high dimensions. Overall, it builds a bridge from classical OT theory to modern statistical and machine-learning methodologies grounded in optimal transport geometry.

Abstract

We present an introduction to the field of statistical optimal transport, based on lectures given at École d'Été de Probabilités de Saint-Flour XLIX.
Paper Structure (6 sections, 2 theorems, 24 equations, 3 figures)

This paper contains 6 sections, 2 theorems, 24 equations, 3 figures.

Table of Contents

  1. Optimal transport
  2. The optimal transport problem
  3. The Monge and Kantorovich problems
  4. Couplings
  5. Discrete optimal transport
  6. Wasserstein distances

Key Result

proposition thmcounterproposition

Let $\mu, \nu$ be two probability measures on $\mathbb{R}^d$. The set $\varGamma_{\mu, \nu}$ of couplings between $\mu$ and $\nu$ is non-empty, convex, and compact with respect to the topology of weak convergence.

Figures (3)

  • Figure 1: Dependencies between chapters. Solid arrows show prerequisites; dotted arrows indicate references.
  • Figure 2: (Left) Independent coupling of a mixture of two Gaussians. (Right) Deterministic coupling of $X\sim \mathcal{N}(0,1)$ with $Y \sim \chi_1^2$.
  • Figure 3: The bivariate Gaussian coupling \ref{['eq:bivariate_gaussian']} for five different values of $\rho$.

Theorems & Definitions (5)

  • proposition thmcounterproposition
  • proof
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proof