Efficient local search for L_1 and L_2 binary matrix factorization

Article type: Research Article

Authors: Mirisaee, Hamid^{a; *} | Gaussier, Eric^a | Termier, Alexandre^b

Affiliations: [a] University of Grenoble Alps/CNRS, Grenoble, France | [b] University of Rennes I, France

Correspondence: [*] Corresponding author: Hamid Mirisaee, 110 Ave. de la Chimie, BP 53, 38041 Grenoble, France. Tel.: +33 4 56 52 03 20; Fax: +33 4 76 51 42 52; E-mail:[email protected]

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.

Keywords: Binary matrix factorization, local search, heuristic

DOI: 10.3233/IDA-160832

Journal: Intelligent Data Analysis, vol. 20, no. 4, pp. 783-807, 2016