Multichannel extensions of non-negative matrix factorization with complex-valued data

Abstract—This paper presents new formulations and algorithms for multichannel extensions of non-negative matrix factorization (NMF). The formulations employ Hermitian positive semidefinite matrices to represent a multichannel version of non-negative elements. Multichannel Euclidean distance and multichannel Itakura-Saito (IS) divergence are defined based on appropriate statistical models utilizing multivariate complex Gaussian distri-butions. To minimize this distance/divergence, efficient optimiza-tion algorithms in the form of multiplicative updates are derived by using properly designed auxiliary functions. Two methods are proposed for clustering NMF bases according to the estimated spatial property. Convolutive blind source separation (BSS) is performed by the multichannel extensions of NMF with the clus-tering mechanism. Experimental results show that 1) the derived multiplicative update rules exhibited good convergence behavior, and 2) BSS tasks for several music sources with two microphones and three instrumental parts were evaluated successfully. Index Terms—Blind source separation, clustering, convolutive mixture, multichannel, non-negative matrix factorization. I