  In many applications in statistics and machine learning, the availability of
data samples from multiple possibly heterogeneous sources has become
increasingly prevalent. On the other hand, in distributionally robust
optimization, we seek data-driven decisions which perform well under the most
adverse distribution from a nominal distribution constructed from data samples
within a certain discrepancy of probability distributions. However, it remains
unclear how to achieve such distributional robustness in model learning and
estimation when data samples from multiple sources are available. In this work,
we propose constructing the nominal distribution in optimal transport-based
distributionally robust optimization problems through the notion of Wasserstein
barycenter as an aggregation of data samples from multiple sources. Under
specific choices of the loss function, the proposed formulation admits a
tractable reformulation as a finite convex program, with powerful finite-sample
and asymptotic guarantees. As an illustrative example, we demonstrate with the
problem of distributionally robust sparse inverse covariance matrix estimation
for zero-mean Gaussian random vectors that our proposed scheme outperforms
other widely used estimators in both the low- and high-dimensional regimes.
