From Convolutional Sparse Coding To *-NMF Factorization of Time-Frequency Coefficients

Convolutional Dictionary Learning (CDL) is a dictionary learning technique exploiting the translation invariance of elementary signals. In the time-frequency domain, the repetition of elementary frequency patterns can be exploited through the "nonnegative matrix factorization" (NMF) decompositions and extensions, such as semi or complex-NMF, of the spectrogram. We study the links between these two approaches here, and we show in particular that a signal which admits a Convolutive Sparse Coding decomposition admits time-frequency synthesis coefficients that can be decomposed in semi-NMF or complex-NMF. The different approaches are then compared experimentally on synthetic signals.