Matrix Completion with Selective Sampling

Matrix completion is a classical problem in data science wherein one attempts to reconstruct a low-rank matrix while only observing some subset of the entries. Previous authors have phrased this problem as a nuclear norm minimization problem. Almost all previous work assumes no explicit structure of the matrix and uses uniform sampling to decide the observed entries. We suggest methods for selective sampling in the case where we have some knowledge about the structure of the matrix and are allowed to design the observation set.