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.

本文探讨了在流式数据下，从独立同分布的数据采样中估算协方差矩阵的主特征向量的问题，并提出和分析了一种分布式变体方案 D-Krasulina，该方案可以在多个处理节点上分布计算负载以跟上高数据流率。通过对该方案的分析表明，在适当的条件下，D-Krasulina 以次序优化的方式收敛于主特征向量，即在接收到所有节点的 M 个采样后，其估算误差可以是 O (1/M)。最后，通过对合成和真实世界数据的实验验证了 D-Krasulina 和 DM-Krasulina 在高速数据流场景中的收敛行为。