  We consider the problem of recovering an unknown low-rank matrix X with
(possibly) non-orthogonal, effectively sparse rank-1 decomposition from
incomplete and inaccurate measurements y gathered in a linear measurement
process A. We propose a variational formulation that lends itself to
alternating minimization and whose global minimizers provably approximate X
from y up to noise level. Working with a variant of robust injectivity, we
derive reconstruction guarantees for various choices of A including
sub-gaussian, Gaussian rank-1, and heavy-tailed measurements. Numerical
experiments support the validity of our theoretical considerations.
