Entropic Optimal Transport on Countable Spaces: Statistical Theory and Asymptotics

In recent years, the theory of OT has become a fundamental tool in statistical research, e.g. the Wasserstein distance being a prominent method for inferential purposes. The large computational complexity, however, has hindered OT in becoming a routine methodology for the analysis of large scale data sets. This has encouraged the formulation of regularized OT which often turns out to be computationally more accessible. The most prominent proposal is given by entropic regularization (c.f. [1] ) that serves to define an entropic OT distance (EOTD) and Sinkhorn divergence.

In the present study, we derive limit distributions for empirical EOTD (i.e. when data are sampled randomly) between probability measures supported on countable discrete spaces. In particular, we consider a general class of cost functions and state conditions on the probability measures to ensure general weak convergence for empirical EOTD. Furthermore, for bounded cost functions we show that the empirical entropic transport plan itself converges weakly in $\ell^1$-sense to a Gaussian process. The theory generalizes results derived by [2] for finite discrete spaces. Moreover, they complement recent findings by [3] for the empirical EOTD between more general probability measures on $\mathbb{R}^m$ with quadratic cost.

Our approach is based on a sensitivity analysis of necessary and sufficient optimality conditions for the entropic transport plan. We demonstrate possible application for colocalization analysis of protein networks in biology.

This is joint work with Marcel Klatt and Axel Munk.

References:
[1] - Marco Cuturi. Sinkhorn Distances: Lightspeed Computation of Optimal Transport. In Advances in Neural Information Processing Systems 26, pages 2292-2300, 2013
[2] - Marcel Klatt, Carla Tameling, and Axel Munk. Empirical Regularized Optimal Transport: Statistical Theory and Applications. arXiv, 2018
[3] - Gonzalo Mena, and Jonathan Weed. Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem. arXiv, 2019