Entropic uncertainty inequalities on sparse representation

Abstract

In this study, some new entropic inequalities on sparse representation for pairs of bases are investigated. First, the generalised Shannon entropic uncertainty principle and the generalised Renyi entropic uncertainty principle via new derived Hausdorff–Young inequality are proved. These new derived uncertainty principles show that signals cannot have unlimited concentration related to minimum entropies in pairs of bases. Second, the conditions of uniqueness of sparse representation under minimum entropies are given. This study also demonstrates that the entropic greedy-like algorithms can achieve the ‘sparsest’ representation for minimum entropies approximately. Third, the relations between the minimum l 0 solution and minimum entropy are discussed as well. It shows that even if the sparsest representation of l 0-norm is obtained, the entropy cannot always be the minimum and it is possible that the entropy is limited to a interval. These new derived inequalities will be primarily to contribute to a better understanding of sparse representation in the sense of limited entropic uncertainty bounds. Finally, the experiments are shown to verify the authors’ ideas.