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. Our 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 is inspired by a popular algorithm from the data mining community called PROXIMUS: it adopts the same recursive partitioning approach while extending to missing data. The algorithm relies upon rankone approximations of incomplete binary matrices, and we propose a linear programming (LP) approach for solving this subproblem. We also prove a 2approximation result for the LP approach which holds for any level of subsampling and for any subsampling pattern. Our numerical experiments show that TBMC outperforms existing methods on recommender systems arising in the context of real datasets.

Binary Matrix Factorisation via Column Generation

“…(Shi, Wang, and Shi 2014) provide a rounding based 2-approximation for 1-BMF by using an observation about the vertices of the polytope corresponding to the LP relaxation of an integer program. In (Beckerleg and Thompson 2020) a modification of the Proximus framework is explored using the approach of (Shen, Ji, and Ye 2009) to compute 1-BMF at each step.…”