Relative Pairwise Relationship Constrained Non-negative Matrix Factorisation

TL;DR

This paper introduces RPR-NMF, a novel matrix factorisation method that incorporates relative pairwise relationship constraints to improve clustering and recommendation tasks, with proven convergence and superior empirical performance.

Contribution

It proposes a new NMF variant that models pairwise relationships using triplet constraints and develops an efficient algorithm with convergence guarantees.

Findings

RPR-NMF achieves closer approximation to data.

It satisfies a high proportion of pairwise constraints.

It outperforms existing NMF methods in experiments.

Abstract

Non-negative Matrix Factorisation (NMF) has been extensively used in machine learning and data analytics applications. Most existing variations of NMF only consider how each row/column vector of factorised matrices should be shaped, and ignore the relationship among pairwise rows or columns. In many cases, such pairwise relationship enables better factorisation, for example, image clustering and recommender systems. In this paper, we propose an algorithm named, Relative Pairwise Relationship constrained Non-negative Matrix Factorisation (RPR-NMF), which places constraints over relative pairwise distances amongst features by imposing penalties in a triplet form. Two distance measures, squared Euclidean distance and Symmetric divergence, are used, and exponential and hinge loss penalties are adopted for the two measures respectively. It is well known that the so-called "multiplicative update rules" result in a much faster convergence than gradient descend for matrix factorisation. However, applying such update rules to RPR-NMF and also proving its convergence is not straightforward. Thus, we use reasonable approximations to relax the complexity brought by the penalties, which are practically verified. Experiments on both synthetic datasets and real datasets demonstrate that our algorithms have advantages on gaining close approximation, satisfying a high proportion of expected constraints, and achieving superior performance compared with other algorithms.