Semi-discrete optimal transport: hardness, regularization and numerical solution

Abstract Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems ubiquitous in statistics, machine learning computer vision, however, this perception has not yet received theoretical justification. To fill gap, we prove that computing measure supported on two points Lebesgue standard hypercube is already $$\#$$ # P-hard. This insight prompts us seek approximate solutions for semi-discrete problems. We thus perturb underlying transportation cost with an additive disturbance governed by ambiguous distribution, introduce distributionally robust dual problem whose objective function smoothed most adverse distributions from within given ambiguity set. further show smoothing equivalent regularizing primal function, identify several sets give rise known new regularization schemes. As byproduct, discover intimate relation choice models traditionally studied psychology economics. solve regularized efficiently, use stochastic gradient descent algorithm imprecise oracles. A convergence analysis reveals improves best guarantee entropic regularizers.