Robust Sensing of Low-Rank Matrices with Non-Orthogonal Sparse Decomposition

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.