  In this thesis, we consider the Wasserstein barycenter problem of discrete
probability measures from computational and statistical sides. The statistical
focus is estimating the sample size of measures necessary to calculate an
approximation for Fr\'echet mean (barycenter) of a probability distribution
with a given precision. For empirical risk minimization approaches, the
question of the regularization is also studied together with proposing a new
regularization which contributes to the better complexity bounds in comparison
with quadratic regularization. The computational focus is developing algorithms
for calculating Wasserstein barycenters: both primal and dual algorithms which
can be executed in a decentralized manner. The motivation for dual approaches
is closed-forms for the dual formulation of entropy-regularized Wasserstein
distances and their derivatives, whereas the primal formulation has closed-form
expression only in some cases, e.g., for Gaussian measures. Moreover, the dual
oracle returning the gradient of the dual representation for
entropy-regularized Wasserstein distance can be computed for a cheaper price in
comparison with the primal oracle returning the gradient of the
entropy-regularized Wasserstein distance. The number of dual oracle calls, in
this case, will also be less, i.e., the square root of the number of primal
oracle calls. This explains the successful application of the first-order dual
approaches for the Wasserstein barycenter problem.
