Expectation Maximization (EM) is the standard method to learn Gaussian mixtures. Yet its classic, centralized form is often infeasible, due to privacy concerns and computational and communication bottlenecks. Prior work dealt with data distributed by examples, horizontal partitioning, but we lack a counterpart for data scattered by features, an increasingly common scheme (e.g. user profiling with data from multiple entities). To fill this gap, we provide an EM-based algorithm to fit Gaussian mixtures to Vertically Partitioned data (VP-EM). In federated learning setups, our algorithm matches the centralized EM fitting of Gaussian mixtures constrained to a subspace. In arbitrary communication graphs, consensus averaging allows VP-EM to run on large peer-to-peer networks as an EM approximation. This mismatch comes from consensus error only, which vanishes exponentially fast with the number of consensus rounds. We demonstrate VP-EM on various topologies for both synthetic and real data, evaluating its approximation of centralized EM and seeing that it outperforms the available benchmark.
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