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Temporal extensions of nonnegative matrix factorization

Abstract : 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 NMF. The dependencies between consecutive frames of the spectrogram can be imposed either on the basis matrix B or on the activations H (introduced in Chapter 8). The former case is known as the convolutive NMF, reviewed in Section 1.1. In this case, the repeating patterns within data are represented with multidimensional bases which are not vectors anymore, but functions that can span an arbitrary number of dimensions (e.g., time and frequency). The other case consists in imposing temporal structure on the activations H, 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 models that will be discussed in Section 1.2. Special cases will be reviewed in subsequent sections. Continuous models are addressed in Sections 1.3 and 1.4, while Section 1.5 reviews models that involve a discrete latent state variable. Sections 1.6 and 1.7 provide quantitative and qualitative comparisons of the proposed methods, while Section 1.8 summarizes. This chapter is an extended version of the review paper (Smaragdis et al., 2014). In this chapter, we will denote by b V the nonnegative spectral data, with columnŝ v(n) and coe cientsv fn. In most cases, b V is either the magnitude spectrogram |X| or the power spectrogram |X| 2 , i.e.,v fn = |x(n, f)| or |x(n, f)| 2. We will also denote V = BH, with coe cients v fn. Note that traditionally the NMF literature instead denotes the data by V and the approximate factorization by b V. However the chosen notation is here consistent with the convention used in this book, where variable with a hat denote statistics (observed quantities) and variables without a hat denote model parameters.
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Cédric Févotte, Paris Smaragdis, Nasser Mohammadiha, Gautham Jean Mysore. Temporal extensions of nonnegative matrix factorization. Audio Source Separation and Speech Enhancement, 2018. ⟨hal-02376817⟩

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