Multichannel Extensions of Non-Negative Matrix Factorization With Complex-Valued Data

Abstract

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 distributions. To minimize this distance/divergence, efficient optimization 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 clustering 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.

Probabilistic integration of diffuse noise suppression and dereverberation

“…Here, · F denotes the Frobenius norm, and E[y t y H t ] is practically replaced by an empirical estimate using short-term data around time t. On the other hand, the likelihood maximization in Section 2.3 is equivalent to minimization of a multichannel Itakura-Saito divergence [16][17][18]:…”

“…Owing to the scale invariance, the Itakura-Saito divergence can more properly measure the discrepancy between audio spectra, which have a logarithmic nature [16].…”

“…Indeed, let F = I MT , and (16) reduces to the standard multichannel Wiener filter (4). Hence, (16) is the cascade of denoising with the generalized multichannel Wiener filter, and dereverberation with the dereverberation operator F H .…”

Stereophonic music separation based on non-negative tensor factorization with cepstrum regularization

“…(7) and Eq. (9). By partially differentiating the auxiliary function with respective parameters and setting the resulting derivatives to 0, we obtain update rules for each parameter as follows:…”

“…This extension has been well studied as a sound source separation technique using microphone array. For instance, Multichannel NMF (MNMF) was proposed [8], [9] to make it possible to effectively use the spatial information by modeling inter-channel phase information as well as the amplitude or power spectrograms of the individual sound source signals. Furthermore, as an approach to the BSS in an overdetermined condition, Independent Low-Rank Matrix Analysis (ILRMA) [10] was proposed to explicitly use a physical constraint on the spatial mixing process.…”