From Convolutionnal dictionnary learning to Time-frequency NMF

This dissertation establishes a unified theoretical framework connecting two major low-rank decomposition paradigms: Convolutional Dictionary Learning (CDL) and Non-negative Matrix Factorization (NMF) in the time–frequency domain. We show that a univariate convolutional model can be rigorously interpreted as a rank-1 NMF factorization of time–frequency coefficients — and, under explicit conditions, the converse also holds.

This equivalence naturally extends to higher-order convolutive factorizations through the Low-Rank Time-Frequency Synthesis (LRTFS) model, and to multivariate signals via Non-negative Tensor Factorization (NTF).

From a methodological perspective, we introduce the CDLFirst strategy, which performs convolutional decomposition directly in sensor space before solving the inverse problem. While formally equivalent to a source-space factorization, this approach leverages the reduced sensor dimensionality to substantially decrease computational cost. We further demonstrate that initializing CDL algorithms with NMF significantly accelerates convergence and improves temporal atom recovery.

Although initially motivated by applications in M/EEG analysis for epileptogenic spike modeling, the proposed framework applies broadly to settings where temporal convolutional structures interact with low-rank spectral representations, including audio processing, vibration analysis, and multivariate time series. Experiments on real M/EEG signals additionally reveal benefits such as automatic spike detection, morphological clustering, and learning interpretable spatial maps that facilitate inverse solving.