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Hierarchical and tree-like data sets arise in many applications, includinglanguage processing, graph data mining, phylogeny and genomics. It is knownthat tree-like data cannot be embedded into Euclidean spaces of finitedimension with small distortion. This problem can be mitigated through the useof hyperbolic spaces. When such data also has to be processed in a distributedand privatized setting, it becomes necessary to work with new federatedlearning methods tailored to hyperbolic spaces. As an initial step towards thedevelopment of the field of federated learning in hyperbolic spaces, we proposethe first known approach to federated classification in hyperbolic spaces. Ourcontributions are as follows. First, we develop distributed versions of convexSVM classifiers for Poincar\'e discs. In this setting, the information conveyedfrom clients to the global classifier are convex hulls of clusters present inindividual client data. Second, to avoid label switching issues, we introduce anumber-theoretic approach for label recovery based on the so-called integer$B_h$ sequences. Third, we compute the complexity of the convex hulls inhyperbolic spaces to assess the extent of data leakage; at the same time, inorder to limit communication cost for the hulls, we propose a new quantizationmethod for the Poincar\'e disc coupled with Reed-Solomon-like encoding. Fourth,at the server level, we introduce a new approach for aggregating convex hullsof the clients based on balanced graph partitioning. We test our method on acollection of diverse data sets, including hierarchical single-cell RNA-seqdata from different patients distributed across different repositories thathave stringent privacy constraints. The classification accuracy of our methodis up to $\sim 11\%$ better than its Euclidean counterpart, demonstrating theimportance of privacy-preserving learning in hyperbolic spaces.

Federated Classification in Hyperbolic Spaces via Secure Aggregation of Convex Hulls