Phase transition in PCA with missing data: Reduced signal-to-noise ratio, not sample size!
This addresses a fundamental issue in data analysis for researchers and practitioners using PCA with incomplete datasets, offering a novel theoretical insight that could improve estimation methods.
The authors tackled the problem of how missing data affects learning signal structures via principal component analysis, finding that missing data effectively reduces the signal-to-noise ratio rather than sample size, with theory predicting and simulations confirming a phase transition in learning curves.
How does missing data affect our ability to learn signal structures? It has been shown that learning signal structure in terms of principal components is dependent on the ratio of sample size and dimensionality and that a critical number of observations is needed before learning starts (Biehl and Mietzner, 1993). Here we generalize this analysis to include missing data. Probabilistic principal component analysis is regularly used for estimating signal structures in datasets with missing data. Our analytic result suggests that the effect of missing data is to effectively reduce signal-to-noise ratio rather than - as generally believed - to reduce sample size. The theory predicts a phase transition in the learning curves and this is indeed found both in simulation data and in real datasets.