Data embedding and prediction by sparse tropical matrix factorization

TL;DR

This paper introduces Sparse Tropical Matrix Factorization (STMF), a novel non-linear matrix factorization method using tropical semiring, which improves missing data estimation and pattern recovery, especially in biological datasets, outperforming traditional NMF.

Contribution

The paper presents the first application of tropical semiring to sparse data in matrix factorization, enhancing modeling of complex relations and outlier fitting compared to NMF.

Findings

STMF achieves higher correlation than NMF on synthetic data.

STMF outperforms NMF on six of nine gene expression datasets.

STMF better fits extreme values and distributions than NMF.

Abstract

Matrix factorization methods are linear models, with limited capability to model complex relations. In our work, we use tropical semiring to introduce non-linearity into matrix factorization models. We propose a method called Sparse Tropical Matrix Factorization (STMF) for the estimation of missing (unknown) values. We evaluate the efficiency of the STMF method on both synthetic data and biological data in the form of gene expression measurements downloaded from The Cancer Genome Atlas (TCGA) database. Tests on unique synthetic data showed that STMF approximation achieves a higher correlation than non-negative matrix factorization (NMF), which is unable to recover patterns effectively. On real data, STMF outperforms NMF on six out of nine gene expression datasets. While NMF assumes normal distribution and tends toward the mean value, STMF can better fit to extreme values and distributions. STMF is the first work that uses tropical semiring on sparse data. We show that in certain cases semirings are useful because they consider the structure, which is different and simpler to understand than it is with standard linear algebra.