[Paper Review] Graphical and uniform consistency of estimated optimal transport plans

Johan Segers|arXiv (Cornell University)|Aug 4, 2022

Fuzzy Systems and Optimization4 citations

TL;DR

This paper establishes graphical and uniform consistency of estimated optimal transport plans between probability measures in Euclidean space by treating maximal cyclically monotone mappings as closed sets in a product space and applying random set theory in the Fell topology. The key contribution is a general consistency result under weak moment conditions, ensuring local uniform convergence of transport plans and enabling valid inference for multivariate ranks and quantiles based on optimal transport.

ABSTRACT

A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based on the identification of such a mapping with a closed subset of a Cartesian product of Euclidean spaces and leveraging tools from random set theory. Weak convergence in the appropriate Fell space together with the maximal cyclical monotonicity then automatically yields local uniform convergence of the associated mappings. Viewing such mappings as optimal transport plans between probability measures with respect to the squared Euclidean distance as cost function yields consistency results for notions of multivariate ranks and quantiles based on optimal transport, notably the empirical center-outward distribution and quantile functions.

Motivation & Objective

• To establish graphical and uniform consistency of estimated optimal transport plans between probability measures in Euclidean space.
• To provide a general consistency framework for multivariate ranks and quantiles derived from optimal transport, especially in nonparametric statistics.
• To extend Glivenko–Cantelli-type results for transport maps under minimal moment assumptions, avoiding restrictive density or support conditions.
• To unify the convergence of transport plans via the theory of maximal monotone mappings and set convergence in the Fell topology.
• To support the theoretical validity of center-outward distribution and quantile functions in high-dimensional, distribution-free inference.

Proposed method

• Models optimal transport plans as maximal cyclically monotone mappings, identified as closed subsets in the product space $\mathbb{R}^d \times \mathbb{R}^d$.
• Applies tools from random set theory to analyze convergence of these closed sets under the Fell topology.
• Uses weak convergence in the Fell topology combined with maximal cyclical monotonicity to derive local uniform convergence of the mappings.
• Leverages Prohorov's theorem and weak compactness to ensure pre-compactness of coupling measures and their supports.
• Employs the Knott–Smith optimality criterion and Rockafellar’s theorem to characterize optimal transport plans as subdifferentials of convex functions.
• Applies the criterion from Alberti and Ambrosio (1999) to prove equality of mappings on dense subsets, ensuring graphical convergence.

Experimental results

Research questions

• RQ1 Under what conditions does the estimated optimal transport plan converge graphically and uniformly to the true transport plan?
• RQ2 Can uniform consistency of multivariate rank and quantile functions be established without strong assumptions on densities or supports?
• RQ3 How does the Fell topology facilitate the analysis of convergence for multivalued transport mappings?
• RQ4 To what extent can the convergence of transport plans be guaranteed under weak moment conditions?
• RQ5 What is the role of maximal cyclical monotonicity in ensuring the uniqueness and stability of the limiting transport plan?

Key findings

• The set of all cyclically monotone couplings $\varPi_{\mathrm{cm}}(P,Q)$ is weakly closed, ensuring stability under weak convergence of measures.
• Under weak convergence of $P_n \to P$ and $Q_n \to Q$, the sequence of optimal transport plans $T_n = \partial\psi_n$ converges graphically to $T = \partial\psi$ on any compact set $V$ contained in the interior of the support of $P$.
• Any accumulation point of the sequence $T_n$ is a maximal monotone mapping containing $\operatorname{spt}\pi$, and all such limits agree on $V$, ensuring graphical convergence.
• Local uniform convergence of the transport maps is guaranteed by the combination of Fell topology convergence and maximal cyclical monotonicity.
• The result holds under minimal assumptions—only finite second moments are required for the limiting measures, and no density or support regularity is needed.
• The framework supports the theoretical validity of empirical center-outward quantile and rank functions, enabling distribution-free inference in multivariate statistics.