  Distributed Principal Component Analysis (PCA) has been studied to deal with
the case when data are stored across multiple machines and communication cost
or privacy concerns prohibit the computation of PCA in a central location.
However, the sub-Gaussian assumption in the related literature is restrictive
in real application where outliers or heavy-tailed data are common in areas
such as finance and macroeconomic. In this article, we propose a distributed
algorithm for estimating the principle eigenspaces without any moment
constraint on the underlying distribution. We study the problem under the
elliptical family framework and adopt the sample multivariate Kendall'tau
matrix to extract eigenspace estimators from all sub-machines, which can be
viewed as points in the Grassman manifold. We then find the "center" of these
points as the final distributed estimator of the principal eigenspace. We
investigate the bias and variance for the distributed estimator and derive its
convergence rate which depends on the effective rank and eigengap of the
scatter matrix, and the number of submachines. We show that the distributed
estimator performs as if we have full access of whole data. Simulation studies
show that the distributed algorithm performs comparably with the existing one
for light-tailed data, while showing great advantage for heavy-tailed data. We
also extend our algorithm to the distributed learning of elliptical factor
models and verify its empirical usefulness through real application to a
macroeconomic dataset.

Computation