Incremental Aggregated Riemannian Gradient Method for Distributed PCA

We consider the problem of distributed principal component analysis (PCA) where the data samples are dispersed across different agents. Despite the rich literature on this problem under various specific settings, there is still a lack of efficient algorithms that are amenable to decentralized and asynchronous implementations. In this paper, we extend the incremental aggregated gradient (IAG) method in convex optimization to the nonconvex PCA problems based on an Riemannian gradient-type method named IARG-PCA. The IARG-PCA method admits low per-iteration computational and communication cost and can be readily implemented in a decentralized and asynchronous manner. Moreover, we show that the IARG-PCA method converges linearly to the leading eigenvector of the sample covariance of the whole dataset with a constant step size. The iteration complexity coincides with the best-known result of the IAG method in terms of the linear dependence on the number of agents. Meanwhile, the communication complexity is much lower than the state-of-the-art decentralized PCA algorithms if the eigengap of the sample covariance is moderate. Numerical experiments on synthetic and real datasets show that our IARG-PCA method exhibits substantially lower communication cost and comparable computational cost compared with other existing algorithms.