Relative entropy based uncertainty principles for graph signals

•First, the relative entropy based uncertainty principle in terms of graph vector signal is derived for the first time and it shows the relation between the new bound and the parameters: the length n of the vector v, the min non-zero correlation coefficients ς1 and ς2, and the max correlation coefficients ζ, ξ1 and ξ2. At the same time, the special cases are involved as well.•Second, the relative entropy based uncertainty principle in terms of graph matrix signal is derived for the first time as well and it shows the relation between the new bound and the parameters: the length n of the vector v in matrix eigendecomposition, the number of eigenvalues of matrix: K, the min non-zero correlation coefficients ς1 and ς2, and the max correlation coefficients ζ, ξ1 and ξ2, the min non-zero eigenvalue λmin. In addition, some special cases are shown as well.•Third, in order to avoid the infinitely great quantity, two new combined uncertainty relations are demonstrated at the same time according to the possible case that the given reference signal's energy distribution is not appropriate and fully different from that of the client signal, we still can obtain the limited values of entropy.•Finally, some numerical experiments and certain application are given in great details including theoretical analysis to show the efficiency of these proposed uncertainty relations. In physical quantum mechanics, the uncertainty principle in presence of quantum memory [Berta M, Christandl M, Colbeck R,et al., Nature Physics] can reach much lower bound, which has resulted in a huge breakthrough in quantum mechanics. Inspired by this idea, this paper would propose some novel uncertainty relations in terms of relative entropy for signal representation and time-frequency resolution analysis. On one hand, the relative entropy measures the distinguishability between the known (priori) basis and the client basis, which implies that we have partial “memory” of the client basis so that the uncertainty bounds become sharper in some cases. On the other hand, in some cases, if the reference basis along with nearly the same energy distribution could be given, then the uncertainty bound would tend to zero, as shows that there is no uncertainty any longer. These novel uncertainty relationships with sharper bounds would give us the potential advantages over the classical counterpart. In addition, the detailed comparison with classical Shannon entropy based uncertainty principle has been addressed