We study the problem of distributed estimation of the leading singular vectors for a collection of matrices with shared invariant subspaces. In particular we consider an algorithm that first estimates the projection matrices corresponding to the leading singular vectors for each individual matrix, then computes the average of the projection matrices, and finally returns the leading eigenvectors of the sample averages. We show that the algorithm, when applied to (1) parameters estimation for a collection of independent edge random graphs with shared singular vectors but possibly heterogeneous edge probabilities or (2) distributed PCA for independent sub-Gaussian random vectors with spiked covariance structure, yields estimates whose row-wise fluctuations are normally distributed around the rows of the true singular vectors. Leveraging these results we also consider a two-sample test for the null hypothesis that a pair of random graphs have the same edge probabilities and we present a test statistic whose limiting distribution converges to a central (resp. non-central) $\chi^2$ under the null (resp. local alternative) hypothesis.

翻译：我们研究具有共享不变子空间的矩阵集合前导奇异向量的分布式估计问题。具体考虑一种算法：首先估计每个矩阵对应前导奇异向量的投影矩阵，然后计算投影矩阵的均值，最后返回样本均值的若干前导特征向量。我们证明，将该算法应用于：（1）共享奇异向量但边概率可能异质的独立边随机图集合的参数估计，或（2）具有尖峰协方差结构的独立次高斯随机向量的分布式PCA时，所得估计的行向波动在真实奇异向量各行附近呈正态分布。基于这些结果，我们进一步考虑一个两样本检验，其原假设为一对随机图具有相同边概率。我们给出一个检验统计量，其极限分布在原假设下收敛至中心$\chi^2$分布，在局部备择假设下收敛至非中心$\chi^2$分布。