  We study optimal transport (OT) problem for probability measures supported on
a tree metric space. It is known that such OT problem (i.e., tree-Wasserstein
(TW)) admits a closed-form expression, but depends fundamentally on the
underlying tree structure over supports of input measures. In practice, the
given tree structure may be, however, perturbed due to noisy or adversarial
measurements. To mitigate this issue, we follow the max-min robust OT approach
which considers the maximal possible distances between two input measures over
an uncertainty set of tree metrics. In general, this approach is hard to
compute, even for measures supported in one-dimensional space, due to its
non-convexity and non-smoothness which hinders its practical applications,
especially for large-scale settings. In this work, we propose novel uncertainty
sets of tree metrics from the lens of edge deletion/addition which covers a
diversity of tree structures in an elegant framework. Consequently, by building
upon the proposed uncertainty sets, and leveraging the tree structure over
supports, we show that the robust OT also admits a closed-form expression for a
fast computation as its counterpart standard OT (i.e., TW). Furthermore, we
demonstrate that the robust OT satisfies the metric property and is negative
definite. We then exploit its negative definiteness to propose positive
definite kernels and test them in several simulations on various real-world
datasets on document classification and topological data analysis.
