# Blind Audio Source Separation with Minimum-Volume Beta-Divergence NMF

## TL;DR

This paper introduces a novel NMF-based model with a volume penalty for blind audio source separation, demonstrating improved interpretability and automatic source number estimation under noiseless conditions.

## Contribution

The paper proposes a new NMF model with a volume penalty term, providing theoretical guarantees for source identification and automatic model order selection.

## Key findings

- More interpretable separation results compared to standard NMF.
- Effective source recovery even when the number of sources is overestimated.
- Automatic zeroing of sources in overestimated scenarios.

## Abstract

Considering a mixed signal composed of various audio sources and recorded with a single microphone, we consider on this paper the blind audio source separation problem which consists in isolating and extracting each of the sources. To perform this task, nonnegative matrix factorization (NMF) based on the Kullback-Leibler and Itakura-Saito $\beta$-divergences is a standard and state-of-the-art technique that uses the time-frequency representation of the signal. We present a new NMF model better suited for this task. It is based on the minimization of $\beta$-divergences along with a penalty term that promotes the columns of the dictionary matrix to have a small volume. Under some mild assumptions and in noiseless conditions, we prove that this model is provably able to identify the sources. In order to solve this problem, we propose multiplicative updates whose derivations are based on the standard majorization-minimization framework. We show on several numerical experiments that our new model is able to obtain more interpretable results than standard NMF models. Moreover, we show that it is able to recover the sources even when the number of sources present into the mixed signal is overestimated. In fact, our model automatically sets sources to zero in this situation, hence performs model order selection automatically.