The fractional Kullback-Leibler divergence

The Kullback-Leibler divergence or relative entropy is generalised by deriving its fractional form. The conventional Kullback-Leibler divergence as well as other formulations emerge as special cases. It is shown that the fractional divergence encapsulates different relative entropy states via the manipulation of the fractional order and for this reason it is the evolution equation for relative entropy. The fractional Kullback-Leibler divergence establishes mathematical dualities with other divergences or distance metrics. The fractional-order can be characterised as a distance metric between divergences or relative entropy states. Generalised asymptotic divergences and densities are derived that are mixtures of known approaches.

Automated summary

The Kullback-Leibler divergence is generalized into its fractional form in this study, showing that the fractional divergence can capture different relative entropy states through manipulation of the fractional order. It serves as the evolution equation for relative entropy and establishes mathematical dualities with other divergences or distance metrics. The fractional order can be characterized as a distance metric between divergences or relative entropy states, leading to the derivation of generalized asymptotic divergences and densities that are mixtures of known approaches.