Lower bounds for the low-rank matrix approximation

Low-rank matrix recovery is an active topic drawing the attention of many researchers. It addresses the problem of approximating the observed data matrix by an unknown low-rank matrix. Suppose that A is a low-rank matrix approximation of D, where D and A are m× n matrices. Based on a useful decomposition of D^†− A^†, for the unitarily invariant norm ∥ ⋅ ∥ , when ∥ D∥ ≥ ∥ A∥ and ∥ D∥ ≤ ∥ A∥ , two sharp lower bounds of D− A are derived respectively. The presented simulations and applications demonstrate our results when the approximation matrix A is low-rank and the perturbation matrix is sparse.