[Paper Review] Limit Distributions and Sensitivity Analysis for Entropic Optimal Transport on Countable Spaces

Shayan Hundrieser, Marcel Klatt+1 more|arXiv (Cornell University)|Apr 30, 2021

Geometric Analysis and Curvature Flows48 references4 citations

TL;DR

This paper establishes limit distributions for empirical entropic optimal transport on countable spaces, proving weak convergence of the transport plan to a centered Gaussian process and asymptotic normality of the optimal value. The analysis relies on sensitivity analysis via dual formulation and optimality conditions, yielding weighted Borisov-Dudley-Durst conditions with weights tied to the ground cost and entropic penalty.

ABSTRACT

For probability measures supported on countable spaces we derive limit distributions for empirical entropic optimal transport quantities. In particular, we prove that the corresponding plan converges weakly to a centered Gaussian process. Furthermore, its optimal value is shown to be asymptotically normal. The results are valid for a large class of ground cost functions and generalize recently obtained limit laws for empirical entropic optimal transport quantities on finite spaces. Our proofs are based on a sensitivity analysis with respect to a weighted $\ell^1ehBnorm relying on the dual formulation of entropic optimal transport as well as necessary and sufficient optimality conditions for the entropic transport plan. This can be used to derive weak convergence of the empirical entropic optimal transport plan and value that results in weighted Borisov-Dudley-Durst conditions on the underlying probability measures. The weights are linked to an exponential penalty term for dual entropic optimal transport and the underlying ground cost function under consideration. Finally, statistical applications, such as bootstrap, are discussed.

Motivation & Objective

• To extend limit laws for empirical entropic optimal transport from finite to countable state spaces.
• To establish weak convergence of the empirical entropic transport plan to a centered Gaussian process.
• To prove asymptotic normality of the empirical optimal transport value under general ground cost functions.
• To develop a sensitivity analysis framework using weighted ℓ¹ norms and dual formulations.
• To derive conditions—weighted Borisov-Dudley-Durst—under which convergence holds, linking weights to cost and entropic penalty.

Proposed method

• Utilizes the dual formulation of entropic optimal transport to analyze sensitivity with respect to empirical measures.
• Applies necessary and sufficient optimality conditions for the entropic transport plan to derive convergence behavior.
• Employs a weighted ℓ¹ norm with weights derived from the exponential penalty term and ground cost function.
• Establishes weak convergence of the empirical transport plan by verifying conditions akin to Borisov-Dudley-Durst.
• Derives asymptotic normality of the optimal value using the same sensitivity framework and dual structure.
• Discusses statistical applications such as bootstrap inference based on the derived limit laws.

Experimental results

Research questions

• RQ1 How do empirical entropic optimal transport plans behave asymptotically on countably infinite spaces?
• RQ2 What is the limiting distribution of the empirical optimal transport value in the countable setting?
• RQ3 Can sensitivity analysis with weighted ℓ¹ norms be used to derive convergence conditions for entropic OT?
• RQ4 How do the weights in the convergence conditions relate to the ground cost and entropic regularization?
• RQ5 What are the implications of the derived limit laws for statistical inference, such as bootstrap procedures?

Key findings

• The empirical entropic transport plan converges weakly to a centered Gaussian process on countable spaces.
• The empirical optimal transport value is asymptotically normal under general ground cost functions.
• The convergence is governed by weighted Borisov-Dudley-Durst conditions, with weights derived from the ground cost and entropic penalty.
• The sensitivity analysis framework relies on the dual formulation and optimality conditions of entropic OT.
• The results generalize prior finite-space limit laws to the countable case.
• Statistical applications, including bootstrap, are feasible due to the derived asymptotic normality and convergence structure.