  Optimal transport (OT) is a powerful geometric tool for comparing two
distributions and has been employed in various machine learning applications.
In this work, we propose a novel OT formulation that takes feature correlations
into account while learning the transport plan between two distributions. We
model the feature-feature relationship via a symmetric positive semi-definite
Mahalanobis metric in the OT cost function. For a certain class of regularizers
on the metric, we show that the optimization strategy can be considerably
simplified by exploiting the problem structure. For high-dimensional data, we
additionally propose suitable low-dimensional modeling of the Mahalanobis
metric. Overall, we view the resulting optimization problem as a non-linear OT
problem, which we solve using the Frank-Wolfe algorithm. Empirical results on
the discriminative learning setting, such as tag prediction and multi-class
classification, illustrate the good performance of our approach.
