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