We study the efficiency of greedy algorithms with regard to redundant dictionaries in Hilbert spaces. We obtain upper estimates for the errors of the Pure GreedyAlgorithm and the Orthogonal GreedyAlgorithm in terms of the best m-term approximations. We call such estimates the Lebesgue-type inequalities. We prove the Lebesgue-type inequalities for dictionaries with special structure. We assume t...

This letter considers the problem of dictionary learning for sparse signal representation whose atoms have low mutual coherence. To learn such dictionaries, at each step, we first update the dictionary using the method of optimal directions (MOD) and then apply a dictionary rank shrinkage step to decrease its mutual coherence. In the rank shrinkage step, we first compute a rank 1 decomposition ...

Recently, compressed sensing has been widely applied to various areas such as signal processing, machine learning, and pattern recognition. To find the sparse representation of a vector w.r.t. a dictionary, an l1 minimization problem, which is convex, is usually solved in order to overcome the computational difficulty. However, to guarantee that the l1 minimizer is close to the sparsest solutio...

This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is viable in this context, but also that accurate recovery is possib...

The Precautionary Principle has been an increasingly important principle in international treaties since the 1980s. Through varying formulations, it states that when an activity can lead to a catastrophe for human health or the environment, measures should be taken to prevent it even if the cause-and-effect relationship is not fully established scientifically. The Precautionary Principle has be...

Sparse representation, which uses dictionary atoms to reconstruct input vectors, has been studied intensively in recent years. A proper dictionary is a key for the success of sparse representation. In this paper, an active dictionary learning (ADL) method is introduced, in which classification error and reconstruction error are considered as the active learning criteria in selection of the atom...