This paper focuses on the problem of unsupervised alignment of hierarchical\ndata such as ontologies or lexical databases. This is a problem that appears\nacross areas, from natural language processing to bioinformatics, and is\ntypically solved by appeal to outside knowledge bases and label-textual\nsimilarity. In contrast, we approach the problem from a purely geometric\nperspective: given only a vector-space representation of the items in the two\nhierarchies, we seek to infer correspondences across them. Our work derives\nfrom and interweaves hyperbolic-space representations for hierarchical data, on\none hand, and unsupervised word-alignment methods, on the other. We first\nprovide a set of negative results showing how and why Euclidean methods fail in\nthis hyperbolic setting. We then propose a novel approach based on optimal\ntransport over hyperbolic spaces, and show that it outperforms standard\nembedding alignment techniques in various experiments on cross-lingual WordNet\nalignment and ontology matching tasks.\n