Fast Tree Variants of Gromov Wasserstein

Gromov-Wasserstein (GW) is a powerful tool to compare probability measures whose supports are in different metric spaces. GW suffers however from a computational drawback since it requires to solve a complex non-convex quadratic program. We consider in this work a specific family of ground metrics, namely\textit {tree metrics} for a space of supports of each probability measure in GW. By leveraging a tree structure, we propose to use\textit {flows} from a root to each support to represent a probability measure whose supports are in a tree metric space. We consequently propose a novel tree variant of GW, namely flow-based tree GW (\FlowTGW), by matching the flows of the probability measures. We then show that\FlowTGW~ shares a similar structure as a univariate optimal transport distance. Therefore,\FlowTGW~ is fast for computation and can scale up for large-scale applications. In order to further explore tree structures, we propose another tree variant of GW, namely depth-based tree GW (\DepthTGW), by aligning the flows of the probability measures hierarchically along each depth level of the tree structures. Theoretically, we prove that both\FlowTGW~ and\DepthTGW~ are pseudo-distances. Moreover, we also derive tree-sliced variants, computed by averaging the corresponding tree variants of GW using random tree metrics, built adaptively in spaces of supports. Finally, we test our proposed discrepancies against other baselines on some benchmark tasks.