  We consider the problem of recovering an unknown low-rank matrix X with
(possibly) non-orthogonal, effectively sparse rank-1 decomposition from
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 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.
