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.

翻译：我们提出了一种新的相对熵推广，适用于和大于1的非负向量。在纯组合设置中（不涉及概率考量），我们证明了在定义凸多面体的线性约束存在时，该广义相对熵会出现集中现象，并精确量化了该集中性。我们还给出了概率形式化表述，并将集中性结果扩展至该情形。此外，我们对先前工作进行了若干简化与改进，特别是在优化问题的对偶化、关于$\ell_{\infty}$距离的集中性以及广义KL散度的关联方面。我们的若干结果适用于一般紧凸集（不必是多面体）。