Robust Non-negative Matrix Factorization with β-Divergence for Speech Separation

Abstract

This paper addresses the problem of unsupervised speech separation based on robust non‐negative matrix factorization (RNMF) with β‐divergence, when neither speech nor noise training data is available beforehand. We propose a robust version of non‐negative matrix factorization, inspired by the recently developed sparse and low‐rank decomposition, in which the data matrix is decomposed into the sum of a low‐rank matrix and a sparse matrix. Efficient multiplicative update rules to minimize the β‐divergence‐based cost function are derived. A convolutional extension of the proposed algorithm is also proposed, which considers the time dependency of the non‐negative noise bases. Experimental speech separation results show that the proposed convolutional RNMF successfully separates the repeating time‐varying spectral structures from the magnitude spectrum of the mixture, and does so without any prior training.

Robust Hierarchical Learning for Non-Negative Matrix Factorization With Outliers

“…The second group of techniques for RNMF is to incorporate an extra additive term to accommodate outliers [22]- [26]. In this setting, the observation is reformulated as V ≈ WH +S.…”

“…Most elements in S are zeros with a few outliers. Some previous studies have enforced the outliers to be non-negative for particular applications such as hyper-spectral unmixing [22], speech separation [26], speech enhancement [24], etc. Without loss of generality, the outliers here can include both positive and negative elements.…”

“…The MAP estimate of the outliers associated with the doubleexponential prior is identical to the convex optimization problem listed below, arg min (26) This is a Least Absolute Shrinkage and Selection Operator (LASSO) problem which can be solved efficiently via the soft-threshold operator denoted as (27) where , 0) and v is the threshold parameter.…”

“…By revisiting (26), (27) and (29), it is obvious that we are essentially using hyper-parameters c and d to control the regularization parameter of the 1 -norm penalty term, which is not straightforward. A more simple and straightforward alternative is to alternate c and d by introducing a hyper-parameter γ to control the sparse matrix directly, which is in line with the cases in [7], [18], and [22].…”

Orthogonal nonnegative matrix tri-factorization based on Tweedie distributions

“…This assumption is familiar with a gross summation of the nonnegative values and associated with various types of real-world nonnegative data (e.g., precipitation, insurance, and purchase volume data) (Ohnishi and Dunn 2007;Smyth and Jørgensen 2002). From an NMF parameter estimation aspect, this assumption is related to robust estimation in the presence of outliers that have extremely large positive values (Li et al 2017;Virtanen et al 2015;Carabias-Orti et al 2013;Weninger and Schuller 2012;Févotte et al 2009;Virtanen 2007).…”