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Matrix factorization methods are linear models, with limited capability tomodel complex relations. In our work, we use tropical semiring to introducenon-linearity into matrix factorization models. We propose a method calledSparse Tropical Matrix Factorization (STMF) for the estimation of missing(unknown) values. We evaluate the efficiency of the STMF method on bothsynthetic data and biological data in the form of gene expression measurementsdownloaded from The Cancer Genome Atlas (TCGA) database. Tests on uniquesynthetic data showed that STMF approximation achieves a higher correlationthan non-negative matrix factorization (NMF), which is unable to recoverpatterns effectively. On real data, STMF outperforms NMF on six out of ninegene expression datasets. While NMF assumes normal distribution and tendstoward 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 thatin certain cases semirings are useful because they consider the structure,which is different and simpler to understand than it is with standard linearalgebra.

Data embedding and prediction by sparse tropical matrix factorization