Generalized Fast Multichannel Nonnegative Matrix Factorization Based on Gaussian Scale Mixtures for Blind Source Separation

Abstract

This paper describes heavy-tailed extensions of a state-of-the-art versatile blind source separation method called fast multichannel nonnegative matrix factorization (FastMNMF) from a unified point of view. The common way of deriving such an extension is to replace the multivariate complex Gaussian distribution in the likelihood function with its heavy-tailed generalization, e.g., the multivariate complex Student's t and leptokurtic generalized Gaussian distributions, and tailor-make the corresponding parameter optimization algorithm. Using a wider class of heavy-tailed distributions called a Gaussian scale mixture (GSM), i.e., a mixture of Gaussian distributions whose variances are perturbed by positive random scalars called impulse variables, we propose GSM-FastMNMF and develop an expectationmaximization algorithm that works even when the probability density function of the impulse variables have no analytical expressions. We show that existing heavy-tailed FastMNMF extensions are instances of GSM-FastMNMF and derive a new instance based on the generalized hyperbolic distribution that include the normal-inverse Gaussian, Student's t, and Gaussian distributions as the special cases. Our experiments show that the normalinverse Gaussian FastMNMF outperforms the state-of-the-art FastMNMF extensions and ILRMA model in speech enhancement and separation in terms of the signal-to-distortion ratio.

Source recovery by analytical maximization of phase-shifted kurtosis for the mixtures of noisy and noiseless signals

“…If we can give a well determined form to the expression (7) by taking it as an objective function then we can also determine its parameters in order to obtain the adequate separation angle.…”

“…Expression (7) always has a positive value, and at the angle of separation this function must be at its maximum value since the angle of separation leads to statistical independence. And according to the geometric concept (See Fig.…”

“…The objective in solving the BSS problem is to recover a set of signals called source signals from a set of received signals called mixtures without prior knowledge either of the nature of the source signals or of the mechanism by which the mixing system (propagation medium) mixes the signals. BSS is a very important discipline and it is involved in many research areas such as Wireless Communication [5], Acoustics [6][7][8], Seismic signal analysis [9], Biomedical field [10] ...etc. Blindly and via the BSS, the principle of the solution is based on hypotheses concerning the mixing system as well as hypotheses on the nature of the source signals.…”

Source recovery by analytical maximization of phase-shifted kurtosis for the mixtures of noisy and noiseless signals

“…By exhaustive research we can obtain an approximation of the adequate rotating angle where we can also note that the form of the expression ( 7) is similar to the variation of the phase-shifted cosine function of period �𝑇𝑇 = π 2 �. Therefore, we can express (7) as follows:…”