Estimation With Low-Rank Time–Frequency Synthesis Models

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 us in turn to design more sophisticated multilayer representations that can efficiently capture diverse forms of structure. Additionally, the generative modeling allows us 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.