A divide-and-conquer algorithm for binary matrix completion

Abstract

We propose a practical algorithm for low rank matrix completion for matrices with binary entries which obtains explicit binary factors and show it performs well at the recommender task on real world datasets. The algorithm, which we call TBMC (Tiling for Binary Matrix Completion), gives interpretable output in the form of binary factors which represent a decomposition of the matrix into tiles. Our approach extends a popular algorithm from the data mining community, PROXIMUS, to missing data, applying the same recursive partitioning approach. The algorithm relies upon rank-one approximations of incomplete binary matrices, and we propose a linear programming (LP) approach for solving this subproblem. We also prove a 2-approximation result for the LP approach which holds for any subsampling pattern, and show that TBMC exactly solves the rank-k prediction task for a underlying block-diagonal tiling structure with geometrically decreasing tile sizes, providing the ratio between successive tiles is less than 1/2. Numerical experiments show that TBMC outperforms existing methods on real datasets.