  Many state-of-the-art signal decomposition techniques rely on a low-rank
factorization of a time-frequency (t-f) transform. In particular, nonnegative
matrix factorization (NMF) of the spectrogram has been considered in many audio
applications. This is an analysis approach in the sense that the factorization
is applied to the squared magnitude of the analysis coefficients returned by
the t-f transform. In this paper we instead propose a synthesis approach, where
low-rankness is imposed to the synthesis coefficients of the data signal over a
given t-f dictionary (such as a Gabor frame). As such we offer a novel modeling
paradigm that bridges t-f synthesis modeling and traditional analysis-based NMF
approaches. The proposed generative model allows in turn to design more
sophisticated multi-layer representations that can efficiently capture diverse
forms of structure. Additionally, the generative modeling allows to exploit t-f
low-rankness for compressive sensing. We present efficient iterative shrinkage
algorithms to perform estimation in the proposed models and illustrate the
capabilities of the new modeling paradigm over audio signal processing
examples.
