  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.
