2020/02/20 by Marcel Nutz, Ruodu Wang, Nutz, Marcel +1 · 1 citation
Mathematics · #49N05 #62G10 #93E20 #FOS: Mathematics #Geometric and Algebraic Topology #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2002.08717
openalex publication_date 2020/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We introduce a constrained optimal transport problem where origins x can only be transported to destinations y≥ x. Our statistical motivation is to describe the sharp upper bound for the variance of the treatment effect Y-X given marginals when the effect is monotone, or Y≥ X. We thus focus on supermodular costs (or submodular rewards) and introduce a coupling P* that is optimal for all such costs and yields the sharp bound. This coupling admits manifold characterizations -- geometric, order-theoretic, as optimal transport, through the cdf, and via the transport kernel -- that explain its structure and imply useful bounds. When the first marginal is atomless, P* is concentrated on the graphs of two maps which can be described in terms of the marginals, the second map arising due to the binding constraint.