"Convergence analysis for dictionary learning under a signal model with non-homogeneous distribution of supports and coefficient amplitudes"Kühmeier, Morris-LucaWe show convergence for two alternating minimisation algorithms for dictionary learning under mild conditions. To be precise we prove convergence of the Method of Optimal Directions (MOD) and the algorithm for Online Dictionary Learning (ODL) for data models with non-uniform distribution of the supports of sparse coefficients in combination with non-homogeneous distribution of the coefficient amplitudes. The innovation lies in including coefficients with non-homogeneous sizes, which is a generalization of the results in [1], in which the coefficient amplitudes are uniformly distributed. We prove that a well-behaved initial dictionary contracts to the generating dictionary with geometric convergence rate, if either their distance is not larger than 1/ log(K) or if it is assured that each component of the initial dictionary is associated with exactly one element of the generating dictionary. [1] S. Ruetz and K. Schnass. Convergence of alternating minimisation algorithms for dictionary learning. arXiv:2304.01768, 2023. |
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