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 $m \times n$ matrices. Based on a useful decomposition of $D^{\dagger} - A^{\dagger}$ , for the unitarily invariant norm $\|\cdot\|$ , when $\|D\|\geq\|A\| $ and $\|D\|\leq\|A\|$ , 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.