On the Identifiability of Overcomplete Dictionaries via the Minimisation Principle Underlying K-SVD

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This article gives theoretical insights into the performance of K-SVD, a dictionary learning algorithm that has gained significant popularity in practical applications. The particular question studied here is when a dictionary Φ∈ Rᵈ × ᵏ can be recovered as local minimum of the minimisation criterion underlying K-SVD from a set of N training signals yₙ =Φxₙ. A theoretical analysis of the problem leads to two types of identifiability results assuming the training signals are generated from a tight frame with coefficients drawn from a random symmetric distribution. First, asymptotic results showing, that in expectation the generating dictionary can be recovered exactly as a local minimum of the K-SVD criterion if the coefficient distribution exhibits sufficient decay. Second, based on the asymptotic results it is demonstrated that given a finite number of training samples N, such that N/ N = O(K³d), except with probability O(N-Kd) there is a local minimum of the K-SVD criterion within distance O(KN⁻¹/⁴) to the generating dictionary.