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Matrix Completion via Sparse Factorization Solved by Accelerated Proximal Alternating Linearized Minimization
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DOI：10.1109/TBDATA.2018.2871476.png)
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Abstract
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Classical matrix completion methods are not effective in recovering missing entries of data drawn from multiple subspaces because the matrices are often of high-rank. Recently a few advanced matrix completion methods were proposed to solve the problem but they are not scalable to large matrices and big data problems. This paper proposes a sparse factorization method for matrix completion on multiple-subspace data. The method factorizes the given incomplete matrix into a dense matrix and a sparse matrix, while the factorization errors of the observed entries are minimized. To solve the optimization problem, an accelerated proximal alternating linearized minimization (APALM) algorithm is proposed. As a non-trivial task owing to the alternation, linearization, nonconvexity, and extrapolation, the convergence of APALM is proved. APALM can solve a large class of optimization problems such as matrix factorization with nonsmooth regularizations. In addition, we show that, to recover an $m\times n$mxn matrix consisting of data drawn from $k$k subspaces of dimension $r_0$r0, the number of observed entries required in our matrix completion method is $O(nr_0\,\log k\,\log n)$O(nr0logklogn) while that in conventional methods is $O(nr_0k\,\log n)$O(nr0klogn), which theoretically proves the superiority of our method on multiple-subspace data and high-rank matrices. The proposed matrix completion method is compared with state-of-the-art on synthetic data and real collaborative filtering problems. The experimental results corroborate that the proposed method can handle large matrices efficiently and provide high recovery accuracy.
Keywords:
Matrix completion
sparse factorization
low-rank
accelerated proximal alternating linearized minimization
nonconvex optimization
collaborative filtering
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