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arXiv:1502.01871

Shannon and Renyi Entropy of Wavelets

H. M. de Oliveira

Published Feb 6, 2015• 3 citations
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Abstract

This paper reports a new reading for wavelets, which is based on the classical 'De Broglie' principle. The wave-particle duality principle is adapted to wavelets. Every continuous basic wavelet is associated with a proper probability density, allowing defining the Shannon entropy of a wavelet. Further entropy definitions are considered, such as Jumarie or Renyi entropy of wavelets. We proved that any wavelet of the same family has the same Shannon entropy of its mother wavelet. Finally, the Shannon entropy for a few standard wavelet families is determined.

DOI
10.48550/arxiv.1502.01871

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This paper reports a new reading for wavelets, which is based on the classical 'De Broglie' principle. The wave-particle duality principle is adapted to wavelets. Every continuous basic wavelet is associated with a proper probability density, allowing defining the Shannon entropy of a wavelet. Further entropy definitions are considered, such as Jumarie or Renyi entropy of wavelets. We proved that any wavelet of the same family has the same Shannon entropy of its mother wavelet. Finally, the Shannon entropy for a few standard wavelet families is determined.
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The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation. We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
The Rényi entropy of general order unifies the well-known Shannon entropy with several other entropy notions, like the min-entropy or collision entropy. In contrast to the Shannon entropy, there seems to be no commonly accepted definition for the conditional Rényi entropy: several versions have been proposed and used in the literature. In this paper, we reconsider the definition for the conditional Rényi entropy of general order as proposed by Arimoto in the seventies. We show that this particular notion satisfies several natural properties. In particular, we show that it satisfies monotonicity under conditioning, meaning that conditioning can only reduce the entropy, and (a weak form of) chain rule, which implies that the decrease in entropy due to conditioning is bounded by the number of bits one conditions on. None of the other suggestions for the conditional Rényi entropy satisfies both these properties. Finally, we show a natural interpretation of the conditional Rényi entropy in terms of (unconditional) Rényi divergence, and we show consistency with a recently proposed notion of conditional Rényi entropy in the quantum setting.