A Riemannian gossip approach to subspace learning on Grassmann manifold

TL;DR

This paper introduces a decentralized gossip-based method for subspace learning on the Grassmann manifold, enabling multiple agents to collaboratively learn a global subspace while preserving privacy and reducing communication costs.

Contribution

It proposes a novel cost function and algorithm for decentralized subspace learning on the Grassmann manifold, combining local learning with asymptotic consensus among agents.

Findings

Effective in matrix completion tasks

Achieves consensus on global subspace

Scalable and parallelizable approach

Abstract

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.