# Matrix tri-factorization over the tropical semiring

arXiv: https://arxiv.org/abs/2305.06624 · HTML: https://mlsanity.com/paper/2305.06624/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 31 references that are in the map.
- First posted on arXiv 11 May 2023.

## Builds on (most cited first, 5 of 5)

- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Data Fusion by Matrix Factorization](https://mlsanity.com/paper/1307.0803/) (2013)
- [Latitude: A Model for Mixed Linear-Tropical Matrix Factorization](https://mlsanity.com/paper/1801.06136/) (2018)
- [FastSTMF: Efficient tropical matrix factorization algorithm for sparse data](https://mlsanity.com/paper/2205.06619/) (2022)
- [Algorithms for Approximate Subtropical Matrix Factorization](https://mlsanity.com/paper/1707.08872/) (2017)

## Built on by (most cited first, 1 of 1)

- [Nonnegative Matrix Factorization in Dimensionality Reduction: A Survey](https://mlsanity.com/paper/2405.03615/) (2024)

## Abstract (arXiv)

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.