Student's T nonnegative matrix factorization and positive semidefinite tensor factorization for single-channel audio source separation

This paper presents a robust variant of nonnegative matrix factorization (NMF) based on complex Student's t distributions (t-NMF) for source separation of single-channel audio signals. The Itakura-Saito divergence NMF (Gaussian NMF) is justified for this purpose under an assumption that the complex spectra of source signals and those of the mixture signal are complex Gaussian distributed (the additiv-ity of power spectra holds). In fact, however, the source spectra are often heavy-tailed distributed. When the source spectra are complex Cauchy distributed, for example, the mixture spectra are also complex Cauchy distributed (the additivity of amplitude spectra holds). Using the complex t distribution that includes the complex Gaussian and Cauchy distributions as its special cases, we propose t-NMF as a unified extension of Gaussian NMF and Cauchy NMF. Furthermore, we propose the corresponding variant of positive semidefinite tensor factorization based on multivariate complex t distributions (t-PSDTF). The experimental results showed that while t-NMF and t-PSDTF were comparative to Gaussian counterparts in terms of peak performance, they worked much better on average because they are insensitive to initialization and tend to avoid local optima.