Matrix Factorization in Tropical and Mixed Tropical-Linear Algebras

TL;DR

This paper advances matrix factorization techniques within tropical and mixed tropical-linear algebra, proposing improved algorithms and exploring applications in recommendation systems and utility function learning.

Contribution

It introduces an improved algorithm for Tropical Matrix Factorization and a novel formulation for approximate decomposition involving tropical products, with applications in recommendation systems.

Findings

Proposed algorithms effectively avoid local optima in tropical matrix factorization.

Numerical results demonstrate the algorithms' effectiveness.

Application to recommendation systems shows promising results.

Abstract

Matrix Factorization (MF) has found numerous applications in Machine Learning and Data Mining, including collaborative filtering recommendation systems, dimensionality reduction, data visualization, and community detection. Motivated by the recent successes of tropical algebra and geometry in machine learning, we investigate two problems involving matrix factorization over the tropical algebra. For the first problem, Tropical Matrix Factorization (TMF), which has been studied already in the literature, we propose an improved algorithm that avoids many of the local optima. The second formulation considers the approximate decomposition of a given matrix into the product of three matrices where a usual matrix product is followed by a tropical product. This formulation has a very interesting interpretation in terms of the learning of the utility functions of multiple users. We also present numerical results illustrating the effectiveness of the proposed algorithms, as well as an application to recommendation systems with promising results.