  Supervised learning methods with missing data have been extensively studied
not just due to the techniques related to low-rank matrix completion. Also in
unsupervised learning one often relies on imputation methods. As a matter of
fact, missing values induce a bias in various estimators such as the sample
covariance matrix. In the present paper, a convex method for sparse subspace
estimation is extended to the case of missing and corrupted measurements. This
is done by correcting the bias instead of imputing the missing values. The
estimator is then used as an initial value for a nonconvex procedure to improve
the overall statistical performance. The methodological as well as theoretical
frameworks are applied to a wide range of statistical problems. These include
sparse Principal Component Analysis with different types of randomly missing
data and the estimation of eigenvectors of low-rank matrices with missing
values. Finally, the statistical performance is demonstrated on synthetic data.
