In Optimal Transport (OT) on a finite metric space, one defines a distance on the probability simplex that extends the distance on the ground space. The distance is the value of a Linear Programming (LP) problem on the set of non-negative-valued 2-way tables with assigned probability functions as margins. We apply to this case the methodology of moves from Algebraic Statistics (AS) and use it to derive a Monte Carlo Markov Chain (MCMC) solution algorithm.
"Finite space Kantorovich problem with an MCMC of table moves." Electron. J. Statist. 15 (1) 880 - 907, 2021. https://doi.org/10.1214/21-EJS1804