Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization will exactly recover the low rank matrix. In this article, we analyze the scenario in which the non-uniform sampling distribution may or may not not align with the underlying matrix's leverage scores. Here we explore learning the parameters for weighted nuclear norm minimization in terms of the empirical sampling distribution. We provide a sufficiency condition for these learned weights which provide an exact recovery guarantee for weighted nuclear norm minimization. It has been established that a specific choice of weights in terms of the true sampling distribution not only allows for weighted nuclear norm minimization to exactly recover the low rank matrix, but also allows for a quantifiable relaxation in the exact recovery conditions. In this article we extend this quantifiable relaxation in exact recovery conditions for a specific choice of weights defined analogously in terms of the empirical distribution as opposed to the true sampling distribution. To accomplish this we employ a concentration of measure bound and a large deviation bound. We also present numerical evidence for the healthy robustness of the weighted nuclear norm minimization algorithm to the choice of empirically learned weights. These numerical experiments show that for a variety of easily computable empirical weights, weighted nuclear norm minimization outperforms unweighted nuclear norm minimization in the non-uniform sampling regime.
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