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We give the first algorithm for Matrix Completion whose running time andsample complexity is polynomial in the rank of the unknown target matrix,linear in the dimension of the matrix, and logarithmic in the condition numberof the matrix. To the best of our knowledge, all previous algorithms eitherincurred a quadratic dependence on the condition number of the unknown matrixor a quadratic dependence on the dimension of the matrix in the running time.Our algorithm is based on a novel extension of Alternating Minimization whichwe show has theoretical guarantees under standard assumptions even in thepresence of noise.

Fast matrix completion without the condition number