Expectation-Maximization Algorithms for Itakura-Saito Nonnegative Matrix Factorization

Abstract

This paper presents novel expectation-maximization (EM) algorithms for estimating the nonnegative matrix factorization model with Itakura-Saito divergence. Indeed, the common EMbased approach exploits the space-alternating generalized EM (SAGE) variant of EM but it usually performs worse than the conventional multiplicative algorithm. We propose to explore more exhaustively those algorithms, in particular the choice of the methodology (standard EM or SAGE variant) and the latent variable set (full or reduced). We then derive four EM-based algorithms, among which three are novel. Speech separation experiments show that one of those novel algorithms using a standard EM methodology and a reduced set of latent variables outperforms its SAGE variants and competes with the conventional multiplicative algorithm.

Complex ISNMF: A Phase-Aware Model for Monaural Audio Source Separation

“…We remark that if κ = 0, then λ = ρ = 0: therefore, q j,f t = 0 and p j,f t becomes the posterior power of s j,f t , as mentioned in Section III-C1. Then, we recognize in (25) the IS divergence between P j and W j H j , as in the EM algorithm for ISNMF [46]. Consequently, the updates rules ( 36) and ( 37) are similar to those obtained in such a scenario [46], up to an additional power 1/2, which is common when applying the majorizeminimization methodology for estimating ISNMF [44].…”

“…To assess the validity of this update scheme, we applied both procedures (maximization of the exact functional (44) and its approximation (46)) on the learning dataset used in the experimental evaluation (see Section IV-A). The average relative difference between the phases obtained with those two approaches was of approximately .…”