  We study the problem of high-dimensional Principal Component Analysis (PCA)
with missing observations. In simple, homogeneous missingness settings with a
noise level of constant order, we show that an existing inverse-probability
weighted (IPW) estimator of the leading principal components can (nearly)
attain the minimax optimal rate of convergence. However, deeper investigation
reveals both that, particularly in more realistic settings where the
missingness mechanism is heterogeneous, the empirical performance of the IPW
estimator can be unsatisfactory, and moreover that, in the noiseless case, it
fails to provide exact recovery of the principal components. Our main
contribution, then, is to introduce a new method for high-dimensional PCA,
called `primePCA', that is designed to cope with situations where observations
may be missing in a heterogeneous manner. Starting from the IPW estimator,
primePCA iteratively projects the observed entries of the data matrix onto the
column space of our current estimate to impute the missing entries, and then
updates our estimate by computing the leading right singular space of the
imputed data matrix. It turns out that the interaction between the
heterogeneity of missingness and the low-dimensional structure is crucial in
determining the feasibility of the problem. We therefore introduce an
incoherence condition on the principal components and prove that in the
noiseless case, the error of primePCA converges to zero at a geometric rate
when the signal strength is not too small. An important feature of our
theoretical guarantees is that they depend on average, as opposed to
worst-case, properties of the missingness mechanism. Our numerical studies on
both simulated and real data reveal that primePCA exhibits very encouraging
performance across a wide range of scenarios.
