Optimally Weighted PCA for High-Dimensional Heteroscedastic Data

Modern data are increasingly both high-dimensional and heteroscedastic. This paper considers the challenge of estimating underlying principal components from high-dimensional data with noise that is heteroscedastic across samples, i.e., some samples are noisier than others. Such heteroscedasticity naturally arises, e.g., when combining data from diverse sources or sensors. A natural way to account for this heteroscedasticity is to give noisier blocks of samples less weight in PCA by using the leading eigenvectors of a weighted sample covariance matrix. We consider the problem of choosing weights to optimally recover the underlying components. In general, one cannot know these optimal weights since they depend on the underlying components we seek to estimate. However, we show that under some natural statistical assumptions the optimal weights converge to a simple function of the signal and noise variances for high-dimensional data. Surprisingly, the optimal weights are not the inverse noise variance weights commonly used in practice. We demonstrate the theoretical results through numerical simulations and comparisons with existing weighting schemes. Finally, we briefly discuss how estimated signal and noise variances can be used when the true variances are unknown, and we illustrate the optimal weights on real data from astronomy.