Non-negative multiple matrix factorization

Non-negative Matrix Factorization (NMF) is a tra-ditional unsupervised machine learning technique for decomposing a matrix into a set of bases and co-efficients 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 in-cluded in the data is insufficient to decompose. Our key idea is that we can ease this problem if comple-mentary data are available that we could integrate into the estimation of the bases and coefficients. In this paper, we propose a novel matrix factoriza-tion method called Non-negative Multiple Matrix Factorization (NM2F), which utilizes complemen-tary 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 auxil-iary matrices provide information about the bases and coefficients. We formulate NM2F as a gen-eralization of NMF, and then present a parameter estimation procedure derived from the multiplica-tive update rule. We examined NM2F in both syn-thetic and real data experiments. The effect of the auxiliary matrices appeared in the improved NM2F performance. We also confirmed that the bases that NM2F obtained from the real data were intu-itive and reasonable thanks to the non-negative con-straint.