Title: Box-To-Box Transformation for Modeling Joint Hierarchies. Abstract: Learning representations of entities and relations in knowledge graphs is an active area of research, with much emphasis placed on choosing the appropriate geometry to capture tree-like structures. Box embeddings (Vilnis et al., 2018; Li et al., 2019; Dasgupta et al., 2020), which represent concepts as n-dimensional hyperrectangles, are capable of embedding trees by training on a subset of the transitive closure. In Patel et al. (2020), the authors demonstrate that only the transitive reduction is required, and further extend box embeddings to capture joint hierarchies by augmenting the graph with new nodes.  While it is possible to represent joint hierarchies with this method, the parameters for each hierarchy are decoupled, making generalization between hierarchies infeasible. In this work, we introduce a learned box-to-box transformation which respects the geometric structure of the box embeddings.  We demonstrate that this not only improves the capability of modeling cross-hierarchy compositional edges but is also capable of generalizing from a subset of the transitive reduction.