Towards Complex Nonnegative Matrix Factorization with the Beta-Divergence

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.