  This paper considers the problem of estimating the principal eigenvector of a
covariance matrix from independent and identically distributed data samples in
streaming settings. The streaming rate of data in many contemporary
applications can be high enough that a single processor cannot finish an
iteration of existing methods for eigenvector estimation before a new sample
arrives. This paper formulates and analyzes a distributed variant of the
classical Krasulina's method (D-Krasulina) that can keep up with the high
streaming rate of data by distributing the computational load across multiple
processing nodes. The analysis shows that---under appropriate
conditions---D-Krasulina converges to the principal eigenvector in an
order-wise optimal manner; i.e., after receiving $M$ samples across all nodes,
its estimation error can be $O(1/M)$. In order to reduce the network
communication overhead, the paper also develops and analyzes a mini-batch
extension of D-Krasulina, which is termed DM-Krasulina. The analysis of
DM-Krasulina shows that it can also achieve order-optimal estimation error
rates under appropriate conditions, even when some samples have to be discarded
within the network due to communication latency. Finally, experiments are
performed over synthetic and real-world data to validate the convergence
behaviors of D-Krasulina and DM-Krasulina in high-rate streaming settings.
