Considering a mixed signal composed of various audio sources and recorded with a single microphone, we consider in 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.

中文翻译：

---

具有最小音量 Beta 发散 NMF 的盲音频源分离

考虑由各种音频源组成并用单个麦克风记录的混合信号，我们在本文中考虑了盲音频源分离问题，该问题包括隔离和提取每个源。为了执行此任务，基于 Kullback-Leibler 和 Itakura-Saito $\beta$-divergences 的非负矩阵分解 (NMF) 是一种标准的、最先进的技术，它使用信号的时频表示。我们提出了一种更适合此任务的新 NMF 模型。它基于 $\beta$-divergences 的最小化以及一个惩罚项，该惩罚项使字典矩阵的列具有较小的体积。在一些温和的假设和无噪声条件下，我们证明该模型能够证明可以识别来源。为了解决这个问题，我们提出了乘法更新，其推导基于标准的专业化-最小化框架。我们在几个数值实验中表明，我们的新模型能够获得比标准 NMF 模型更具可解释性的结果。此外，我们表明，即使混合信号中存在的源数量被高估，它也能够恢复源。事实上，在这种情况下，我们的模型会自动将源设置为零，因此会自动执行模型顺序选择。我们表明，即使混合信号中存在的源数量被高估，它也能够恢复源。事实上，在这种情况下，我们的模型会自动将源设置为零，因此会自动执行模型顺序选择。我们表明，即使混合信号中存在的源数量被高估，它也能够恢复源。事实上，在这种情况下，我们的模型会自动将源设置为零，因此会自动执行模型顺序选择。