  Optimal transport (OT) theory provides powerful tools to compare probability
measures. However, OT is limited to nonnegative measures having the same mass,
and suffers serious drawbacks about its computation and statistics. This leads
to several proposals of regularized variants of OT in the recent literature. In
this work, we consider an \textit{entropy partial transport} (EPT) problem for
nonnegative measures on a tree having different masses. The EPT is shown to be
equivalent to a standard complete OT problem on a one-node extended tree. We
derive its dual formulation, then leverage this to propose a novel
regularization for EPT which admits fast computation and negative definiteness.
To our knowledge, the proposed regularized EPT is the first approach that
yields a \textit{closed-form} solution among available variants of unbalanced
OT. For practical applications without priori knowledge about the tree
structure for measures, we propose tree-sliced variants of the regularized EPT,
computed by averaging the regularized EPT between these measures using random
tree metrics, built adaptively from support data points. Exploiting the
negative definiteness of our regularized EPT, we introduce a positive definite
kernel, and evaluate it against other baselines on benchmark tasks such as
document classification with word embedding and topological data analysis. In
addition, we empirically demonstrate that our regularization also provides
effective approximations.
