Tropical semiring has proven successful in several research areas, including optimal control, bioinformatics, discrete event systems, or solving a decision problem. In previous studies, a matrix two-factorization algorithm based on the tropical semiring has been applied to investigate bipartite and tripartite networks. Tri-factorization algorithms based on standard linear algebra are used for solving tasks such as data fusion, co-clustering, matrix completion, community detection, and more. However, there is currently no tropical matrix tri-factorization approach, which would allow for the analysis of multipartite networks with a high number of parts. To address this, we propose the triFastSTMF algorithm, which performs tri-factorization over the tropical semiring. We apply it to analyze a four-partition network structure and recover the edge lengths of the network. We show that triFastSTMF performs similarly to Fast-NMTF in terms of approximation and prediction performance when fitted on the whole network. When trained on a specific subnetwork and used to predict the whole network, triFastSTMF outperforms Fast-NMTF by several orders of magnitude smaller error. The robustness of triFastSTMF is due to tropical operations, which are less prone to predict large values compared to standard operations.

翻译：热带半环已在多个研究领域被证明有效，包括最优控制、生物信息学、离散事件系统以及决策问题的求解。先前研究中，一种基于热带半环的矩阵双因子分解算法已被应用于分析二部网络和三部网络。基于标准线性代数的三因子分解算法则用于解决数据融合、共聚类、矩阵补全、社区检测等任务。然而，目前尚缺乏能够分析多部分网络（尤其是高维部分）的热带矩阵三因子分解方法。为此，我们提出triFastSTMF算法，该算法在热带半环上实现三因子分解。我们将其应用于分析四分区网络结构并恢复网络边长度。结果表明，当在全网络上拟合时，triFastSTMF在近似性能和预测性能方面与Fast-NMTF表现相当。当在特定子网络上训练并用于预测全网络时，triFastSTMF的误差比Fast-NMTF低数个数量级。triFastSTMF的鲁棒性源于热带运算相比标准运算更不易预测较大数值的特性。