  In this paper, we develop an algorithm for federated principal component
analysis (PCA) with emphases on both communication efficiency and data privacy.
Generally speaking, federated PCA algorithms based on direct adaptations of
classic iterative methods, such as simultaneous subspace iterations (SSI), are
unable to preserve data privacy, while algorithms based on variable-splitting
and consensus-seeking, such as alternating direction methods of multipliers
(ADMM), lack in communication-efficiency. In this work, we propose a novel
consensus-seeking formulation by equalizing subspaces spanned by splitting
variables instead of equalizing variables themselves, thus greatly relaxing
feasibility restrictions and allowing much faster convergence. Then we develop
an ADMM-like algorithm with several special features to make it practically
efficient, including a low-rank multiplier formula and techniques for treating
subproblems. We establish that the proposed algorithm can better protect data
privacy than classic methods adapted to the federated PCA setting. We derive
convergence results, including a worst-case complexity estimate, for the
proposed ADMM-like algorithm in the presence of the nonlinear equality
constraints. Extensive empirical results are presented to show that the new
algorithm, while enhancing data privacy, requires far fewer rounds of
communication than existing peer algorithms for federated PCA.
