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Matrix tri-factorization over the tropical semiring

Amra Omanović, Polona Oblak, Tomaž Curk

TL;DR

The paper introduces triFastSTMF, the first tropical semiring-based tri-factorization method, enabling four-partition network analysis by decomposing a data matrix $R$ into $R \approx G_1 \otimes S \otimes G_2$. It builds on STMF/FastSTMF and leverages greatest subsolutions to compute the middle factor $S$, providing a robust approach that matches standard methods in approximation while delivering superior predictive accuracy when trained on subnetworks. Through synthetic and real-data experiments, triFastSTMF demonstrates competitive edge with added robustness against overfitting inherent to tropical operations, and establishes a foundation for tropical data fusion. The work offers practical implications for multipartite network analysis and reproducible research via publicly available code and data.

Abstract

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.

Matrix tri-factorization over the tropical semiring

TL;DR

The paper introduces triFastSTMF, the first tropical semiring-based tri-factorization method, enabling four-partition network analysis by decomposing a data matrix into . It builds on STMF/FastSTMF and leverages greatest subsolutions to compute the middle factor , providing a robust approach that matches standard methods in approximation while delivering superior predictive accuracy when trained on subnetworks. Through synthetic and real-data experiments, triFastSTMF demonstrates competitive edge with added robustness against overfitting inherent to tropical operations, and establishes a foundation for tropical data fusion. The work offers practical implications for multipartite network analysis and reproducible research via publicly available code and data.

Abstract

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.
Paper Structure (18 sections, 3 theorems, 23 equations, 8 figures, 3 tables, 4 algorithms)

This paper contains 18 sections, 3 theorems, 23 equations, 8 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Methods
  4. Our contribution
  5. Semirings $(\max,+)$ and $(\min,+)$
  6. Tri-factorization over the tropical semiring
  7. Different aspects of the tri-factorization on networks
  8. Comparison with other strategies
  9. Synthetic data
  10. Real data
  11. Evaluation metrics
  12. Evaluation
  13. Results
  14. Synthetic data
  15. Comparison between the tropical matrix factorization methods
  16. ...and 3 more sections

Key Result

Theorem 1

For any $A \in {\mathbb{R}}_{\max}^{m \times n}$ and $B \in {\mathbb{R}}_{\max}^{m \times p}$ the greatest subsolution of the equation $A \otimes X=B$ is

Figures (8)

  • Figure 1: Schematic diagram of one iteration of the proposed triFastSTMF method for updating factor matrices $G_1, S$ and $G_2$ of the data matrix $R\approx G_1\otimes S\otimes G_2$. Step 1) updates the factor matrix $G_1$ through CFL, while step 2) uses the new $G_1$ to update $G_2$ through CFR. The last step, 3) updates $S$ using Theorem \ref{['thm:gss']} and newly-computed factor matrices $G_1$ and $G_2$. The procedure repeats until convergence.
  • Figure 2: Example of a four-partition network.
  • Figure 3: A real-world network of the daily average frequency of interactions in an ant colony. The strength of the interaction is visualized with the distance between nodes and edge transparency.
  • Figure 4: Comparison of different tropical tri-factorization methods. The median, first and third quartiles of the approximation error in 25 runs on the synthetic random tropical $200 \times 100$ matrix are shown.
  • Figure 5: (a) A synthetic random tropical network $K$ of 100 nodes created by applying the tropical semiring on four sets $A, B, C$ and $D$. The sets $A$ and $D$ are densely connected, following the network construction process. In contrast, sets $B$ and $C$ are less connected. Example of partitioning network $K$, using b) random and c) partially-random partitioning.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Theorem 1: Described by Gaubert and Plus gaubert
  • Lemma 1
  • proof
  • Theorem 2
  • proof