Alternating Minimization is a widely used and empirically successful framework for Matrix Completion and related low-rank optimization problems. 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 while achieving a linear convergence rate. Compared to previous work our results reduce the provable sample complexity 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 when the matrix is exactly low-rank and when it is only close to low-rank in the Frobenius norm. Underlying our work is a new robust convergence analysis of the well-known Subspace Iteration algorithm for computing the dominant singular vectors of a matrix also known as the Power Method. This viewpoint leads to a conceptually simple understanding of Alternating Minimization that we exploit. Additionally, we contribute a new technique for controlling the coherence of intermediate solutions arising in iterative algorithms. These techniques may be of interest beyond their application here.
As an alternative to the nuclear norm approach, Keshavan, Montanari and Oh @cite_3 @cite_27 present two approaches, a spectral approach and an algorithm called OptSpace . The spectral approach roughly corresponds to our initialization procedure and gives similar guarantees. OptSpace requires a stronger incoherence assumption, has larger sample complexity in terms of the condition number, namely @math and requires optimizing over the Grassmanian manifold. However, the requirement on @math achieved by OptSpace can be weaker than ours in the noisy setting. In the exact case, our algorithm has a much faster convergence rate (logarithmic dependence on @math rather than polynomial).
@cite_27 @cite_3
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Given a matrix M of low-rank, we consider the problem of reconstructing it from noisy observations of a small, random subset of its entries. The problem arises in a variety of applications, from collaborative filtering (the 'Netflix problem') to structure-from-motion and positioning. We study a low complexity algorithm introduced by Keshavan, Montanari, and Oh (2010), based on a combination of spectral techniques and manifold optimization, that we call here OPTSPACE. We prove performance guarantees that are order-optimal in a number of circumstances. Let M be an n? × n matrix of rank r, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm, which we call OptSpace, that reconstructs M from |E| = O(rn) observed entries with relative root mean square error 1 2 RMSE ? C(?) (nr |E|)1 2 with probability larger than 1 - 1 n3. Further, if r = O(1) and M is sufficiently unstructured, then OptSpace reconstructs it exactly from |E| = O(n log n) entries with probability larger than 1 - 1 n3. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|r log n), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices.