Towards Complex Nonnegative Matrix Factorization with the Beta-Divergence

Abstract

Complex nonnegative matrix factorization (NMF) is a powerful tool for decomposing audio spectrograms while accounting for some phase information in the time-frequency domain. While its estimation was originally based on the Euclidean distance, in this paper we propose to extend it to any beta-divergence, a family of functions widely used in audio to estimate NMF. To this end, we introduce the beta-divergence in a heuristic fashion within a phase-aware probabilistic model. Estimating this model results in performing an NMF with Itakura-Saito (IS) divergence on a quantity called the phase-corrected posterior power of the sources, which is both phase-dependent and nonnegative-valued. Therefore, we replace IS with the beta-divergence, so that the factorization uses an optimal distortion metric and remains phase-aware. Even though by doing so we loose theoretical convergence guarantees, the resulting algorithm demonstrates its potential for an audio source separation task, where it outperforms previous complex NMFs approaches.

Efficient Noisy Sound-Event Mixture Classification Using Adaptive-Sparse Complex-Valued Matrix Factorization and OvsO SVM

“…The term F(h) in Equation (16) as the function of the Gibbs distribution is essential for simplifying the adaptive optimization of λ. The maximum-likelihood (ML) estimation of λ can be decomposed as follows:…”

“…The complex nonnegative matrix factorization (CMF) spreads the NMF model by combining a sparsity representation with the complex-spectrum domain to improve the audio separability. The CMF can extract the recurrent patterns of the phase estimates and magnitude spectra of constituent signals [16][17][18]. Nevertheless, the CMF lacks the generalized mechanics used for controlling the sparseness of the code.…”

Nonnegative matrix factorization based on complex generative model

“…Many algorithms have been proposed to solve the above-mentioned problem. In this article, a complexvalued extension of NMF (complex NMF: CNMF) [18][19][20] and NMF based on complex generative models [3,[21][22][23] are reviewed.…”

“…Similar to NMF, the iterative update rules for CNMF can be derived by an auxiliary function technique [18]. In recent years, the similarity function in CNMF is generalized to -divergence, which includes the generalized KL divergence and IS divergence [19,20]. As described above, CNMF assumes the additivity of complex-valued spectrogram components and the low rank of the amplitude spectrogram, resulting in an appropriate decomposition model without ignoring phase information.…”

“…the sum of r.v.s can be modeled by the sum of the firstorder expectations (scale parameters) > 0. The complexvalued spectral components c ij;k are assumed to obey (20), and the scale parameters defined in each time-frequency slot correspond to the expectation of amplitude values as ij;k ¼ E½jc ij;k j. Since the complex Cauchy distribution has the stable property, the generative model of the observed spectrum…”