Noisy Matrix Completion Using Alternating Minimization

The task of matrix completion involves estimating the entries of a matrix, M ∈ Rm×n, when a subset, Ω ⊂ {(i, j) : 1 ≤ i ≤ m, 1 ≤ j ≤ n} of the entries are observed. A particular set of low rank models for this task approximate the matrix as a product of two low rank matrices, M̂ = UV T , where U ∈ Rm×k and V ∈ Rn×k and k min{m,n}. A popular algorithm of choice in practice for recovering M from the partially observed matrix using the low rank assumption is alternating least square (ALS) minimization, which involves optimizing over U and V in an alternating manner to minimize the squared error over observed entries while keeping the other factor fixed. Despite being widely experimented in practice, only recently were theoretical guarantees established bounding the error of the matrix estimated from ALS to that of the original matrix M . In this work we extend the results for a noiseless setting and provide the first guarantees for recovery under noise for alternating minimization. We specifically show that for well conditioned matrices corrupted by random noise of bounded Frobenius norm, if the number of observed entries is O ( kn logn ) , then the ALS algorithm recovers the original matrix within an error bound that depends on the norm of the noise matrix. The sample complexity is the same as derived in [7] for the noise–free matrix completion using ALS.