A Fractional Cross-Entropy Based οn Caputo Fractional-Order Derivative

Cross-entropy (CE) is a measure of how different two probability distributions are. It is commonly used in machine learning and information theory to compare the predicted probability distribution with the actual probability distribution. On the other hand, fractional calculus is a branch of mathematical analysis which studies different possible approaches of defining fractional-order integrals and derivatives. In this paper, we have derived a novel generalized fractional CE (FCE). To do so, we have differentiated the CE's generating function (Formula Presented) using a α -order Caputo fractional-order derivative CD (Formula Presented) When the order of differentiation α → 1,we recover the ordinary Shannon's CE, which corresponds to the results from a first-order ordinary differentiation. This allows to calculate the FCE for various values of α ≠ 1 and compare them to the conventional CE (α = 1) to get further insights on its behavior. Some examples and illustrations of the proposed FCE are also presented. © 2023 School of Science, IHU. All rights reserved.