[Paper Review] Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences

Alberto González-Sanz, Jean–Michel Loubes+1 more|arXiv (Cornell University)|Jul 15, 2022

Probabilistic and Robust Engineering Design8 citations

TL;DR

This paper establishes the weak convergence of entropic regularized optimal transport solutions, deriving central limit theorems for Sinkhorn potentials and couplings as Gaussian processes in Hölder spaces, and characterizing the limiting distribution of the Sinkhorn divergence under both null and alternative hypotheses. It resolves a conjecture on the weak limit of the entropic coupling and provides a foundation for statistical inference using entropic optimal transport.

ABSTRACT

This work deals with the asymptotic distribution of both potentials and couplings of entropic regularized optimal transport for compactly supported probabilities in $\R^d$. We first provide the central limit theorem of the Sinkhorn potentials -- the solutions of the dual problem -- as a Gaussian process in $\Cs$. Then we obtain the weak limits of the couplings -- the solutions of the primal problem -- evaluated on integrable functions, proving a conjecture of \cite{ChaosDecom}. In both cases, their limit is a real Gaussian random variable. Finally we consider the weak limit of the entropic Sinkhorn divergence under both assumptions $H_0:\ { m P}={ m Q}$ or $H_1:\ { m P} eq{ m Q}$. Under $H_0$ the limit is a quadratic form applied to a Gaussian process in a Sobolev space, while under $H_1$, the limit is Gaussian. We provide also a different characterisation of the limit under $H_0$ in terms of an infinite sum of an i.i.d. sequence of standard Gaussian random variables. Such results enable statistical inference based on entropic regularized optimal transport.

Motivation & Objective

• To derive the weak limit of the Sinkhorn potentials (dual solutions) in Hölder spaces, showing convergence to a Gaussian process.
• To establish the weak limit of the entropic coupling (primal solution) for integrable test functions, confirming a conjecture by Harchaoui et al. (2020).
• To characterize the asymptotic distribution of the entropic Sinkhorn divergence under both the null hypothesis (P = Q) and alternative (P ≠ Q).
• To provide a new infinite series representation of the limiting distribution under the null hypothesis using i.i.d. standard Gaussian variables.

Proposed method

• Derives a central limit theorem for Sinkhorn potentials in the space $\mathcal{C}^s(\Omega)$, showing they converge weakly to a Gaussian process.
• Uses a functional delta method and Fréchet differentiability of nonlinear operators to analyze the limit of the coupling via the potential's convergence.
• Applies the Donsker property and entropy-based bounds to control the empirical process in Hölder norms.
• Establishes the weak convergence of the entropic coupling evaluated on bounded continuous functions using a stability bound involving the $\mathcal{C}^s$-norm of the potential difference.
• Characterizes the limiting distribution of the Sinkhorn divergence under $H_0$ as a quadratic form of a Gaussian process in a Sobolev space.
• Provides an alternative characterization of the $H_0$ limit as an infinite sum of i.i.d. standard Gaussian variables.

Experimental results

Research questions

• RQ1 What is the weak limit of the Sinkhorn potentials (dual solutions) when empirical measures are used in place of true distributions?
• RQ2 Does the entropic coupling converge weakly to a Gaussian process when evaluated on integrable functions, confirming the conjecture of Harchaoui et al. (2020)?
• RQ3 What is the asymptotic distribution of the entropic Sinkhorn divergence under the null hypothesis $P = Q$?
• RQ4 How does the limiting distribution of the Sinkhorn divergence behave under the alternative hypothesis $P \neq Q$?
• RQ5 Can the null limiting distribution of the Sinkhorn divergence be represented as an infinite sum of i.i.d. standard Gaussian variables?

Key findings

• The Sinkhorn potentials converge weakly to a Gaussian process in $\mathcal{C}^s(\Omega)$, establishing a central limit theorem for the dual solutions.
• The entropic coupling converges weakly to a Gaussian process when evaluated on integrable functions, confirming the conjecture of Harchaoui, Liu, and Pal (2020).
• Under the null hypothesis $P = Q$, the limiting distribution of the Sinkhorn divergence is a quadratic form of a Gaussian process in a Sobolev space.
• Under the alternative hypothesis $P \neq Q$, the limiting distribution of the Sinkhorn divergence is Gaussian.
• The null limiting distribution of the Sinkhorn divergence admits an alternative representation as an infinite sum of i.i.d. standard Gaussian random variables.
• The convergence rates of the potentials and couplings are controlled via stability bounds in Hölder norms, relying on the Fréchet differentiability of nonlinear operators in $\mathcal{C}^s$ spaces.