High dimensional deformed rectangle matrices with applications in matrix denoising

We consider the recovery of a low rank $M \times N$ matrix $S$ from its noisy observation $\tilde{S}$ in two different regimes. Under the assumption that $M$ is comparable to $N$, we propose two optimal estimators for $S$. Our analysis rely on the local behavior of the large dimensional rectangle matrices with finite rank perturbation. We also derive the convergent limits and rates for the singular values and vectors of such matrices.