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.

矩阵分解在机器学习和数据挖掘中有广泛应用，包括协同过滤推荐系统、降维、数据可视化和社区检测。本文基于热带代数和几何在机器学习领域的最新成果，研究了两个涉及矩阵分解的问题。我们提出了一种改进的算法来解决热带矩阵分解问题，并通过将传统矩阵乘法与热带乘法相结合来探讨了近似分解的第二种形式。我们还展示了新算法的有效性并在推荐系统上进行了应用，并取得了有希望的结果。