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Principal component analysis (PCA) is one of the most popular methods fordimension reduction. In light of the rapidly growing large-scale data infederated ecosystems, the traditional PCA method is often not applicable due toprivacy protection considerations and large computational burden. Algorithmswere proposed to lower the computational cost, but few can handle both highdimensionality and massive sample size under the distributed setting. In thispaper, we propose the FAst DIstributed (FADI) PCA method for federated datawhen both the dimension $d$ and the sample size $n$ are ultra-large, bysimultaneously performing parallel computing along $d$ and distributedcomputing along $n$. Specifically, we utilize $L$ parallel copies of$p$-dimensional fast sketches to divide the computing burden along $d$ andaggregate the results distributively along the split samples. We present FADIunder a general framework applicable to multiple statistical problems, andestablish comprehensive theoretical results under the general framework. Weshow that FADI enjoys the same non-asymptotic error rate as the traditional PCAwhen $Lp \ge d$. We also derive inferential results that characterize theasymptotic distribution of FADI, and show a phase-transition phenomenon as $Lp$increases. We perform extensive simulations to show that FADI substantiallyoutperforms the existing methods in computational efficiency while preservingaccuracy, and validate the distributional phase-transition phenomenon throughnumerical experiments. We apply FADI to the 1000 Genomes data to study thepopulation structure.

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