A Generalization of Relative Entropy to Count Vectors and its Concentration Property
<|reference_start|>A Generalization of Relative Entropy to Count Vectors and its Concentration Property: We introduce a new generalization of relative entropy to non-negative vectors with sums $\gt 1$. We show in a purely combinatorial setting, with no probabilistic considerations, that in the presence of linear constraints defining a convex polytope, a concentration phenomenon arises for this generalized relative entropy, and we quantify the concentration precisely. We also present a probabilistic formulation, and extend the concentration results to it. In addition, we provide a number of simplifications and improvements to our previous work, notably in dualizing the optimization problem, in the concentration with respect to $\ell_{\infty}$ distance, and in the relationship to generalized KL-divergence. A number of our results apply to general compact convex sets, not necessarily polyhedral.<|reference_end|>
@article{oikonomou2024a,
    title={A Generalization of Relative Entropy to Count Vectors and its
  Concentration Property},
    author={Kostas N. Oikonomou},
    journal={arXiv preprint arXiv:2404.15867},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.15867},
    primaryClass={cs.IT math.IT}
}