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

Abstract

This paper introduces a phase-aware probabilistic model for audio source separation. Classical source models in the short-time Fourier transform domain use circularly-symmetric Gaussian or Poisson random variables. This is equivalent to assuming that the phase of each source is uniformly distributed, which is not suitable for exploiting the underlying structure of the phase. Drawing on preliminary works, we introduce here a Bayesian anisotropic Gaussian source model in which the phase is no longer uniform. Such a model permits us to favor a phase value that originates from a signal model through a Markov chain prior structure. The variance of the latent variables are structured with nonnegative matrix factorization (NMF). The resulting model is called complex Itakura-Saito NMF (ISNMF) since it generalizes the ISNMF model to the case of non-isotropic variables. It combines the advantages of ISNMF, which uses a distortion measure adapted to audio and yields a set of estimates which preserve the overall energy of the mixture, and of complex NMF, which enables one to account for some phase constraints. We derive a generalized expectation-maximization algorithm to estimate the model parameters. Experiments conducted on a musical source separation task in a semi-informed setting show that the proposed approach outperforms state-of-the-art phaseaware separation techniques.

Expectation-Maximization Algorithms for Itakura-Saito Nonnegative Matrix Factorization

“…In particular, the standard EM algorithm using a reduced set of latent variables provides faster convergence and better separation results than the SAGE algorithm used in the literature [8]. It also compares favorably with the conventional multiplicative algorithm [3], which confirms its potential for estimating more sophisticated NMF models [15].…”

“…This is particularly interesting since in more sophisticated models where the likelihood of the data is not tractable, one cannot apply ML-MUR, and EM-MUR would then fully reveal its potential. For instance, it can be useful for estimating anisotropic Gaussian models with NMF variance [15], or in a multichannel ISNMF framework, where it is common to exploit EM algorithms [16,21].…”

Towards Complex Nonnegative Matrix Factorization with the Beta-Divergence

“…However, this distortion metric is not well adapted to audio [7]. To alleviate this issue, complex NMF has recently been extended to KL [11] and IS [12] divergences.…”

“…However, it has been extended to the KL divergence by using a primal-dual formulation of the optimization problem [11]. It has also recently been extended to the IS divergence [12] by considering a probabilistic framework based on anisotropic Gaussian (AG) distributions [13]. This family of distributions permits us to model the sources with non-uniform phase parameters and to structure the variance parameters by means of an NMF model.…”

“…To achieve this goal, we introduce the beta-divergence in a heuristic fashion within a phase-aware probabilistic framework. We introduce an AG source model that is a simplified version of the original one [12]. Inference of this model results in minimizing the IS divergence between the NMF model and a novel quantity which is the phase-corrected posterior power of the latent sources.…”