Efficient local search for L_1 and L_2 binary matrix factorization

Abstract

Rank K Binary Matrix Factorization (BMF) approximates a binary matrix by the product of two binary matrices of lower rank, K. Several researchers have addressed this problem, focusing on either approximations of rank 1 or higher, using either the L 1 or L 2-norms for measuring the quality of the approximation. The rank 1 problem (for which the L 1 and L 2-norms are equivalent) has been shown to be related to the Integer Linear Programming (ILP) problem. We first show here that the alternating strategy with the L 2-norm, at the core of several methods used to solve BMF, can be reformulated as an Unconstrained Binary Quadratic Programming (UBQP) problem. This reformulation allows us to use local search procedures designed for UBQP in order to improve the solutions of BMF. We then introduce a new local search dedicated to the BMF problem. We show in particular that this solution is in average faster than the previously proposed ones. We then assess its behavior on several collections and methods and show that it significantly improves methods targeting the L 2-norms on all the datasets considered; for the L 1-norm, the improvement is also significant for real, structured datasets and for the BMF problem without the binary reconstruction constraint.