Cauchy nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is an effective and popular low-rank model for nonnegative data. It enjoys a rich background, both from an optimization and probabilistic signal processing viewpoint. In this study, we propose a new cost-function for NMF fitting, which is introduced as arising naturally when adopting a Cauchy process model for audio waveforms. As we recall, this Cauchy process model is the only probabilistic framework known to date that is compatible with having additive magnitude spectrograms for additive independent audio sources. Similarly to the Gaussian power-spectral density, this Cauchy model features time-frequency nonnegative scale parameters, on which an NMF structure may be imposed. The Cauchy cost function we propose is optimal under that model in a maximum likelihood sense. It thus appears as an interesting newcomer in the inventory of useful cost-functions for NMF in audio. We provide multiplicative updates for Cauchy-NMF and show that they give good performance in audio source separation as well as in extracting nonnegative low-rank structures from data buried in very adverse noise.