Structured matrix estimation and completion

We study the problem of matrix estimation and matrix completion under a general framework. This framework includes several important models as special cases such as the Gaussian mixture model, mixed membership model, biclustering model and dictionary learning. We establish the optimal convergence rates in a minimax sense for estimation of the signal matrix under the Frobenius norm and under the spectral norm. As a consequence of our general result we obtain minimax optimal rates of convergence for various special models.