A proof of Hirschman Uncertainty invariance to the order of Rényi entropy for Picket Fence signals, and its relevance in a simplistic recognition experiment

In [1] we developed a new uncertainty measure which incorporates Rényi entropy instead of Shannon entropy. This new uncertainty measure was conjectured to be invariant to the Rényi order α > 0 for the case of the optimizer signals of Hirschman Uncertainty (Picket Fence functions whose lengths are a perfect square).

In this paper, we prove this invariance, and test whether this invariance is predictive in the problem of a simple texture classification for digital images. In the preliminary results, we find that it certainly influences the recognizer performance. Specifically, we find that the recognition performance does not depend significantly on the Rényi parameter α. We hope that these results will be extended to other problems where Rényi entropy is used.