Non-Negative Multiple Matrix Factorization

Non-negative Matrix Factorization (NMF) is a traditional unsupervised machine learning technique for decomposing a matrix into a set of bases and coefficients under the non-negative constraint. NMF with sparse constraints is also known for extracting reasonable components from noisy data. However, NMF tends to give undesired results in the case of highly sparse data, because the information included in the data is insufficient to decompose. Our key idea is that we can ease this problem if complementary data are available that we could integrate into the estimation of the bases and coefficients. In this paper, we propose a novel matrix factorization method called Non-negative Multiple Matrix Factorization (NMMF), which utilizes complementary data as auxiliary matrices that share the row or column indices of the target matrix. The data sparseness is improved by decomposing the target and auxiliary matrices simultaneously, since auxiliary matrices provide information about the bases and coefficients. We formulate NMMF as a generalization of NMF, and then present a parameter estimation procedure derived from the multiplicative update rule. We examined NMMF in both synthetic and real data experiments. The effect of the auxiliary matrices appeared in the improved NMMF performance. We also confirmed that the bases that NMMF obtained from the real data were intuitive and reasonable thanks to the non-negative constraint.