  In this paper, we focus on subspace learning problems on the Grassmann
manifold. Interesting applications in this setting include low-rank matrix
completion and low-dimensional multivariate regression, among others. Motivated
by privacy concerns, we aim to solve such problems in a decentralized setting
where multiple agents have access to (and solve) only a part of the whole
optimization problem. The agents communicate with each other to arrive at a
consensus, i.e., agree on a common quantity, via the gossip protocol.
  We propose a novel cost function for subspace learning on the Grassmann
manifold, which is a weighted sum of several sub-problems (each solved by an
agent) and the communication cost among the agents. The cost function has a
finite sum structure. In the proposed modeling approach, different agents learn
individual local subspace but they achieve asymptotic consensus on the global
learned subspace. The approach is scalable and parallelizable. Numerical
experiments show the efficacy of the proposed decentralized algorithms on
various matrix completion and multivariate regression benchmarks.
