  A low rank matrix X has been contaminated by uniformly distributed noise,
missing values, outliers and corrupt entries. Reconstruction of X from the
singular values and singular vectors of the contaminated matrix Y is a key
problem in machine learning, computer vision and data science. In this paper we
show that common contamination models (including arbitrary combinations of
uniform noise,missing values, outliers and corrupt entries) can be described
efficiently using a single framework. We develop an asymptotically optimal
algorithm that estimates X by manipulation of the singular values of Y , which
applies to any of the contamination models considered. Finally, we find an
explicit signal-to-noise cutoff, below which estimation of X from the singular
value decomposition of Y must fail, in a well-defined sense.
