Alternating minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank optimization problems. Theoretical guarantees for alternating minimization have been hard to come by and are still poorly understood. This is in part because the heuristic is iterative and non-convex in nature. We give a new algorithm based on alternating minimization that provably recovers an unknown low-rank matrix from a random subsample of its entries under a standard incoherence assumption. Our results reduce the sample size requirements of the alternating minimization approach by at least a quartic factor in the rank and the condition number of the unknown matrix. These improvements apply even if the matrix is only close to low-rank in the Frobenius norm. Our algorithm runs in nearly linear time in the dimension of the matrix and, in a broad range of parameters, gives the strongest sample bounds among all subquadratic time algorithms that we are aware of. Underlying our work is a new robust convergence analysis of the well-known Power Method for computing the dominant singular vectors of a matrix. This viewpoint leads to a conceptually simple understanding of alternating minimization. In addition, we contribute a new technique for controlling the coherence of intermediate solutions arising in iterative algorithms based on a smoothed analysis of the QR factorization. These techniques may be of interest beyond their application here.