ℓ₀ based sparse representation

In this monograph, we study the exact ℓ₀ based sparse representation problem. For the classical dictionary learning problem, the solution is obtained by iteratively processing two steps: sparse coding and dictionary updating. However, even the problem associated with sparse coding is non-convex and NP-hard. The method for solving this is to reformulate the problem as mixed integer quadratic programming (MIQP). Then by introducing two optimization techniques, initialization by proximal method and relaxation with augmented contraints, the algorithmis greatly speed up (which is thus called AcMIQP) and applied in image denoising, which shows the good performance. Moreover, the classical problem is extended to learn an incoherent dictionary. For dealing with this problem, AcMIQP or proximal method is used for sparse coding. As for dictionary updating, augmented Lagrangian method (ADMM) and extended proximal alternating linearized minimizing method are combined. This exact ℓ₀ based incoherent dictionary learning is applied in image recovery, which illustrates the improved performance with a lower coherence.