Coherent Matrix Completion

The matrix completion problem concerns the reconstruction of a low-rank matrix from a subset of its entries. In recent years, nuclear norm minimization has emerged as a popular reconstruction method, and there is now a solid theoretical foundation for matrix completion using nuclear norm minimization and related algorithms. However, existing recovery guarantees assume that the revealed entries are uniformly distributed. This severely restricts the types of matrices that can be recovered; they need to be incoherent, or have their mass spread out uniformly across all elements. Here, we show that nuclear norm minimization can in fact recover an arbitrary low-rank matrix from very few entries -- no matter how coherent, or spiky the matrix -- provided the sampling distribution is dependent, in the right way, on the matrix. Within this general framework, incoherent matrix completion from uniformly distributed entries is a special case.