  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.
