ALBERT

Description: We set out to demonstrate that the Rényi entropies are better thought of as operating in a type of non-linear semiring called a positive semifield. We show how the Rényi’s postulates lead to Pap’s g-calculus where the functions carrying out the domain transformation are Rényi’s information function and its inverse. In its turn, Pap’s g-calculus under Rényi’s information function transforms the set of positive reals into a family of semirings where “standard” product has been transformed into sum and “standard” sum into a power-emphasized sum. Consequently, the transformed product has an inverse whence the structure is actually that of a positive semifield. Instances of this construction lead to idempotent analysis and tropical algebra as well as to less exotic structures. We conjecture that this is one of the reasons why tropical algebra procedures, like the Viterbi algorithm of dynamic programming, morphological processing, or neural networks are so successful in computational intelligence applications. But also, why there seem to exist so many computational intelligence procedures to deal with “information” at large.

Description: Entropy, Vol. 20, Pages 498: Assessing Information Transmission in Data Transformations with the Channel Multivariate Entropy Triangle Entropy doi: 10.3390/e20070498 Authors: Francisco J. Valverde-Albacete Carmen Peláez-Moreno Data transformation, e.g., feature transformation and selection, is an integral part of any machine learning procedure. In this paper, we introduce an information-theoretic model and tools to assess the quality of data transformations in machine learning tasks. In an unsupervised fashion, we analyze the transformation of a discrete, multivariate source of information X¯ into a discrete, multivariate sink of information Y¯ related by a distribution PX¯Y¯. The first contribution is a decomposition of the maximal potential entropy of (X¯,Y¯), which we call a balance equation, into its (a) non-transferable, (b) transferable, but not transferred, and (c) transferred parts. Such balance equations can be represented in (de Finetti) entropy diagrams, our second set of contributions. The most important of these, the aggregate channel multivariate entropy triangle, is a visual exploratory tool to assess the effectiveness of multivariate data transformations in transferring information from input to output variables. We also show how these decomposition and balance equations also apply to the entropies of X¯ and Y¯, respectively, and generate entropy triangles for them. As an example, we present the application of these tools to the assessment of information transfer efficiency for Principal Component Analysis and Independent Component Analysis as unsupervised feature transformation and selection procedures in supervised classification tasks.

Description: We introduce a variant of the Rényi entropy definition that aligns it with the well-known Hölder mean: in the new formulation, the r-th order Rényi Entropy is the logarithm of the inverse of the r-th order Hölder mean. This brings about new insights into the relationship of the Rényi entropy to quantities close to it, like the information potential and the partition function of statistical mechanics. We also provide expressions that allow us to calculate the Rényi entropies from the Shannon cross-entropy and the escort probabilities. Finally, we discuss why shifting the Rényi entropy is fruitful in some applications.