Matrix Factorization in Tropical and Mixed Tropical-Linear Algebras
This work addresses matrix factorization challenges in machine learning applications like recommendation systems, though it appears incremental with algorithm improvements and a new hybrid formulation.
The paper tackles matrix factorization problems in tropical algebra by proposing an improved algorithm for Tropical Matrix Factorization that avoids local optima and introducing a novel three-matrix decomposition combining linear and tropical products for utility function learning. Numerical results show effectiveness with promising application to recommendation systems.
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.