Adaptive sparse and dense hybrid representation with nonconvex optimization Sparse representation has been widely used in signal processing, pattern recognition and computer vision etc. Excellent achievements have been made in both theoretical researches and practical applications. However, there are two limitations on the application of classification. One is that sufficient training samples are required for each class, and the other is that samples should be uncorrupted. In order to alleviate above problems, a sparse and dense hybrid representation (SDR) framework has been proposed, where the training dictionary is decomposed into a classspecific dictionary and a nonclassspecific dictionary. SDR puts l1 constraint on the coefficients of classspecific dictionary. Nevertheless, it overemphasizes the sparsity and overlooks the correlation information in classspecific dictionary, which may lead to poor classification results. To overcome this disadvantage, an adaptive sparse and dense hybrid representation with nonconvex optimization (ASDRNO) is proposed in this paper. The trace norm is adopted in classspecific dictionary, which is different from general approaches. By doing so, the dictionary structure becomes adaptive and the representation ability of the dictionary will be improved. Meanwhile, a nonconvex surrogate is used to approximate the rank function in dictionary decomposition in order to avoid a suboptimal solution of the original rank minimization, which can be solved by iteratively reweighted nuclear norm (IRNN) algorithm. Extensive experiments conducted on benchmark data sets have verified the effectiveness and advancement of the proposed algorithm compared with the stateoftheart sparse representation methods.

Variations on the Convolutional Sparse Coding Model Over the past decade, the celebrated sparse representation model has achieved impressive results in various signal and image processing tasks. A convolutional version of this model, termed convolutional sparse coding (CSC), has been recently reintroduced and extensively studied. CSC brings a natural remedy to the limitation of typical sparse enforcing approaches of handling global and highdimensional signals by local, patchbased, processing. While the classic field of sparse representations has been able to cater for the diverse challenges of different signal processing tasks by considering a wide range of problem formulations, almost all available algorithms that deploy the CSC model consider the same   ell _1  ell _2   problem form. As we argue in this paper, this CSC pursuit formulation is also too restrictive as it fails to explicitly exploit some local characteristics of the signal. This work expands the range of formulations for the CSC model by proposing two convex alternatives that merge global norms with local penalties and constraints. The main contribution of this work is the derivation of efficient and provably converging algorithms to solve these new sparse coding formulations.