Incoherent dictionary learning via mixed-integer programming and hybrid augmented Lagrangian

Abstract

During the past decade, the dictionary learning has been a hot topic in sparse representation. With theoretical guarantees, a low-coherence dictionary is demonstrated to optimize the sparsity and improve the accuracy of the performance of signal reconstruction. Two strategies have been investigated to learn incoherent dictionaries: (i) by adding a decorrelation step after the dictionary updating (e.g. INK-SVD), or (ii) by introducing an additive penalty term of the mutual coherence to the general dictionary learning problem. In this paper, we propose a third method, which learns an incoherent dictionary by solving a constrained quadratic programming problem. Therefore, we can learn a dictionary with a prior fixed coherence value, which cannot be realized by the second strategy. Moreover, it updates the dictionary by considering simultaneously the reconstruction error and the incoherence, and thus does not suffer from the performance reduction of the first strategy. The constrained quadratic programming problem is difficult problem due to its non-smoothness and non-convexity. To deal with the problem, a two-step alternating method is used: sparse coding by solving a problem of mixed-integer programming and dictionary updating by the hybrid method of augmented Lagrangian and alternating proximal linearized minimization. Finally, extensive experiments conducted in image denoising demonstrate the relevance of the proposed method, and illustrate the relation between coherence of dictionary and reconstruction quality.