Matrix Completion via Sparse Factorization Solved by Accelerated Proximal Alternating Linearized Minimization

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 <inline-formula><tex-math notation="LaTeX">$m\times n$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>m</mml:mi><mml:mo>×</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="fan-ieq1-2871476.gif"/></alternatives></inline-formula> matrix consisting of data drawn from <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="fan-ieq2-2871476.gif"/></alternatives></inline-formula> subspaces of dimension <inline-formula><tex-math notation="LaTeX">$r_0$</tex-math><alternatives><mml:math><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="fan-ieq3-2871476.gif"/></alternatives></inline-formula>, the number of observed entries required in our matrix completion method is <inline-formula><tex-math notation="LaTeX">$O(nr_0\,\log k\,\log n)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mspace width="0.166667em"/><mml:mo form="prefix">log</mml:mo><mml:mi>k</mml:mi><mml:mspace width="0.166667em"/><mml:mo form="prefix">log</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="fan-ieq4-2871476.gif"/></alternatives></inline-formula> while that in conventional methods is <inline-formula><tex-math notation="LaTeX">$O(nr_0k\,\log n)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>k</mml:mi><mml:mspace width="0.166667em"/><mml:mo form="prefix">log</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="fan-ieq5-2871476.gif"/></alternatives></inline-formula>, 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.