Empirical Bayes PCA in high dimensions
This work addresses the problem of high-dimensional noise in PCA, which is critical for researchers working with large-scale genomic and other high-dimensional datasets.
This paper introduces Empirical Bayes PCA (EB-PCA) to mitigate high-dimensional noise in PCA when data dimensions are comparable to or exceed sample size. EB-PCA achieves Bayes-optimal estimation accuracy in theoretical "spiked" models and significantly improves over standard PCA in simulations and on quantitative benchmarks from the 1000 Genomes Project and International HapMap Project.
When the dimension of data is comparable to or larger than the number of data samples, Principal Components Analysis (PCA) may exhibit problematic high-dimensional noise. In this work, we propose an Empirical Bayes PCA method that reduces this noise by estimating a joint prior distribution for the principal components. EB-PCA is based on the classical Kiefer-Wolfowitz nonparametric MLE for empirical Bayes estimation, distributional results derived from random matrix theory for the sample PCs, and iterative refinement using an Approximate Message Passing (AMP) algorithm. In theoretical "spiked" models, EB-PCA achieves Bayes-optimal estimation accuracy in the same settings as an oracle Bayes AMP procedure that knows the true priors. Empirically, EB-PCA significantly improves over PCA when there is strong prior structure, both in simulation and on quantitative benchmarks constructed from the 1000 Genomes Project and the International HapMap Project. An illustration is presented for analysis of gene expression data obtained by single-cell RNA-seq.