Single-Channel Audio Source Separation with NMF: Divergences, Constraints and Algorithms

Abstract

Spectral decomposition by nonnegative matrix factorisation (NMF) has become state-of-the-art practice in many audio signal processing tasks, such as source separation, enhancement or transcription. This chapter reviews the fundamentals of NMF-based audio decomposition, in unsupervised and informed settings. We formulate NMF as an optimisation problem and discuss the choice of the measure of fit. We present the standard majorisation-minimisation strategy to address optimisation for NMF with common β-divergence, a family of measures of fit that takes the quadratic cost, the generalised Kullback-Leibler divergence and the Itakura-Saito divergence as special cases. We discuss the reconstruction of time-domain components from the spectral factorisation and present common variants of NMFbased spectral decomposition: supervised and informed settings, regularised versions, temporal models.

Identify, Locate and Separate: Audio-Visual Object Extraction in Large Video Collections Using Weak Supervision

“…• Supervised NMF [25]: We use the class labels to train separate dictionaries of size 100 for each music instrument with stochastic mini-batch updates. At test time, depending on the label, the mixture is projected onto the appropriate dictionary for source reconstruction.…”

Using Recurrences in Time and Frequency within U-net Architecture for Speech Enhancement

“…In these methods training data is used to train model whose aim is to reduce even nonstationary noise. Data-driven methods include methods based on nonnegative matrix factorization [3] and deep neural networks (DNNs) [4]. DNNs are used as a nonlinear transformation of a noisy signal to a clean one (mapping-based targets), or to a filtering mask (i.e.…”

Who Said That? a Comparative Study of Non-negative Matrix Factorization Techniques

“…Blind source separation (BSS) is a process in which there is no prior knowledge of the location of the sources or about the sources themselves in the associated audio file. Different techniques have been applied to this problem, for example, independent component analysis (ICA) [1] and non-negative matrix factorization (NMF) [2,3,4,5,6,7,8,9,10,11,12]. According to Mirsamadi et al [13], the disadvantage of ICA is that if this technique is combined with direction of arrival (DoA) it cannot be used to solve the permutation problem for high frequencies when the frequency exceeds the spatial aliasing limit.…”

“…NMF has been applied successfully on short range speech (less than 5 metres) and premixed audio files [3,14,15]. With NMF it is very important to choose the optimal cost function; the Kullback-Leibler cost function is the most popular one to use with NMF.…”

“…With NMF it is very important to choose the optimal cost function; the Kullback-Leibler cost function is the most popular one to use with NMF. Another cost function is the Itakura-Saito diver-gence, which has been successfully used for music analysis to separate out different instruments in an audio track [2,3]. These cost functions are combined with different techniques, for example DoA [15] and convolution, to improve the accuracy of NMF.…”