Connecting the Dots Between CDL and NMF: A Shared Framework for Time-Frequency Decomposition
1mo ago
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IEEEConnecting the Dots Between CDL and NMF: A Shared Framework for Time-Frequency Decompositionieee.orgThis paper investigates the mathematical relationship between Convolutional Dictionary Learning (CDL) and Non-Negative Matrix Factorization (NMF) for unsupervised pattern recovery in signal processing. Both methods are widely used for audio analysis and unsupervised source separation tasks, yet their underlying connections have remained largely unexplored. We establish the equivalence of CDL and NMF in the time-frequency domain by showing that they can be interpreted as low-rank factorizations of time-frequency synthesis coefficients. Main theoretical results demonstrate how signals following a strided convolutional model in the time domain correspond to rank-1 NMF decompositions in the time-frequency domain. We also extend this framework to more complex signals through an estimation procedure based on the Low-Rank Time-Frequency Synthesis (LRTFS) model. We propose a hybrid approach that combines NMF initialization with CDL to improve computational efficiency. Experimental results on synthetic and real-world signals illustrate the comparative performance of these methods, highlighting their strengths and limitations in various signal decomposition tasks.
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