FADI: Fast Distributed Principal Component Analysis With High Accuracy for Large-Scale Federated Data

Principal component analysis (PCA) is one of the most popular methods for dimension reduction. In light of the rapidly growing large-scale data in federated ecosystems, the traditional PCA method is often not applicable due to privacy protection considerations and large computational burden. Algorithms were proposed to lower the computational cost, but few can handle both high dimensionality and massive sample size under the distributed setting. In this paper, we propose the FAst DIstributed (FADI) PCA method for federated data when both the dimension $d$ and the sample size $n$ are ultra-large, by simultaneously performing parallel computing along $d$ and distributed computing along $n$. Specifically, we utilize $L$ parallel copies of $p$-dimensional fast sketches to divide the computing burden along $d$ and aggregate the results distributively along the split samples. We present FADI under a general framework applicable to multiple statistical problems, and establish comprehensive theoretical results under the general framework. We show that FADI enjoys the same non-asymptotic error rate as the traditional PCA when $Lp \ge d$. We also derive inferential results that characterize the asymptotic distribution of FADI, and show a phase-transition phenomenon as $Lp$ increases. We perform extensive simulations to show that FADI substantially outperforms the existing methods in computational efficiency while preserving accuracy, and validate the distributional phase-transition phenomenon through numerical experiments. We apply FADI to the 1000 Genomes data to study the population structure.