Fast and Near-Optimal Matrix Completion via Randomized Basis Pursuit

Motivated by the philosophy and phenomenal success of compressed sensing, the problem of reconstructing a matrix from a sampling of its entries has attracted much attention recently. Such a problem can be viewed as an information-theoretic variant of the well-studied matrix completion problem, and the main objective is to design an efficient algorithm that can reconstruct a matrix by inspecting only a small number of its entries. Although this is an impossible task in general, Cand\`es and co-authors have recently shown that under a so-called incoherence assumption, a rank $r$ $n\times n$ matrix can be reconstructed using semidefinite programming (SDP) after one inspects $O(nr\log^6n)$ of its entries. In this paper we propose an alternative approach that is much more efficient and can reconstruct a larger class of matrices by inspecting a significantly smaller number of the entries. Specifically, we first introduce a class of so-called stable matrices and show that it includes all those that satisfy the incoherence assumption. Then, we propose a randomized basis pursuit (RBP) algorithm and show that it can reconstruct a stable rank $r$ $n\times n$ matrix after inspecting $O(nr\log n)$ of its entries. Our sampling bound is only a logarithmic factor away from the information-theoretic limit and is essentially optimal. Moreover, the runtime of the RBP algorithm is bounded by $O(nr^2\log n+n^2r)$, which compares very favorably with the $\Omega(n^4r^2\log^{12}n)$ runtime of the SDP-based algorithm. Perhaps more importantly, our algorithm will provide an exact reconstruction of the input matrix in polynomial time. By contrast, the SDP-based algorithm can only provide an approximate one in polynomial time.