Tsallis entropy based uncertainty relations on sparse representation for vector and matrix signals

Abstract

In this paper, the Tsallis entropy based novel uncertainty relations on vector signals and matrix signals in terms of sparse representation are deduced for the first time. These new uncertainty bounds are not only related with the entropy parameter, but also related with the vector length, the min non-zero correlation between standard orthogonal basis and the given signals, the max correlation between the two given orthogonal basis and even the eigenvalues along with eigenvectors. Especially, the relationship between uncertainty bounds and matrix eigenvalues is discussed as well, as discloses the new interesting interpretation on sparse representation. In addition, the theoretical analysis and numerical examples have been shown to verify these newly proposed uncertainty principles, e.g., under the case of special parameters of Tsallis entropy, the uncertainty bound reaches its peak value for the sparsest representation of matrix (i.e., only one eigenvalue is not zero). Moreover, various numerical relations between uncertainty bounds and Tsallis entropy parameters are shown in perceptual form, as maybe give us the possible enlightenment or guidance in future sparse representation analysis.