Selecting the number of components in PCA via random signflips

Dimensionality reduction via PCA and factor analysis is an important tool of data analysis. A critical step is selecting the number of components. However, existing methods (such as the scree plot, likelihood ratio, parallel analysis, etc) do not have statistical guarantees in the increasingly common setting where the data are heterogeneous. There each noise entry can have a different distribution. To address this problem, we propose the Signflip Parallel Analysis (Signflip PA) method: it compares data singular values to those of "empirical null" data generated by flipping the sign of each entry randomly with probability one-half. We show that Signflip PA consistently selects factors above the noise level in high-dimensional signal-plus-noise models (including spiked models and factor models) under heterogeneous settings. Here classical parallel analysis is no longer effective. To do this, we rely on recent results in random matrix theory, such as dimension-free operator norm bounds [Latala et al, 2018, Inventiones Mathematicae], and large deviations for the top eigenvalues of nonhomogeneous matrices [Husson, 2020]. We also illustrate that Signflip PA performs well in numerical simulations and on empirical data examples.