Audio Source Separation and Speech Enhancement

9Temporal Extensions of Nonnegative Matrix Factorization

Cédric Févotte Paris Smaragdis Nasser Mohammadiha and Gautham J. Mysore

Temporal continuity is one of the most important features of time series data. Our aim here is to present some of the basic as well as advanced ideas to make use of this information by modeling time dependencies in nonnegative matrix factorization (NMF). The dependencies between consecutive frames of the spectrogram can be imposed either on the basis matrix or on the activations (introduced in Chapter 8). The former case is known as the convolutive NMF, reviewed in Section 9.1. In this case, the repeating patterns within data are represented with multidimensional bases that are not vectors anymore, but functions that can span an arbitrary number of dimensions (e.g., time and frequency). The other case consists of imposing temporal structure on the activations , in line with traditional dynamic models that have been studied extensively in signal processing. Most models considered in the NMF literature can be cast as special cases of a unifying state‐space model, which will be discussed in Section 9.2. Special cases will be reviewed in subsequent sections. Continuous ...