Rank-One Matrix Pursuit for Matrix Completion

Low rank matrix completion has been applied successfully in a wide range of machine learning applications, such as collaborative filtering, image inpainting and Microarray data imputation. However, many existing algorithms are not scalable to large-scale problems, as they involve computing singular value decomposition. In this paper, we present an efficient and scalable algorithm for matrix completion. The key idea is to extend the well-known orthogonal matching pursuit from the vector case to the matrix case. In each iteration, we pursue a rank-one matrix basis generated by the top singular vector pair of the current approximation residual and update the weights for all rank-one matrices obtained up to the current iteration. We further propose a novel weight updating rule to reduce the time and storage complexity, making the proposed algorithm scalable to large matrices. We establish the linear convergence of the proposed algorithm. The fast convergence is achieved due to the proposed construction of matrix bases and the estimation of the weights. We empirically evaluate the proposed algorithm on many real-world large-scale datasets. Results show that our algorithm is much more efficient than state-of-the-art matrix completion algorithms while achieving similar or better prediction performance.