  Distributionally robust optimization (DRO) incorporates robustness against
uncertainty in the specification of probabilistic models. This paper focuses on
mitigating the curse of dimensionality in data-driven DRO problems with optimal
transport ambiguity sets. By exploiting independence across lower-dimensional
components of the uncertainty, we construct structured ambiguity sets that
exhibit a faster shrinkage as the number of collected samples increases. This
narrows down the plausible models of the data-generating distribution and
mitigates the conservativeness that the decisions of DRO problems over such
ambiguity sets may face. We establish statistical guarantees for these
structured ambiguity sets and provide dual reformulations of their associated
DRO problems for a wide range of objective functions. The benefits of the
approach are demonstrated in a numerical example.
