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：本文引入了一种新的相对熵推广形式，适用于元素和大于1的非负向量。我们在纯组合框架下（不涉及概率考量）证明：当存在定义凸多面体的线性约束时，这种广义相对熵会出现集中现象，并对其集中程度进行了精确刻画。我们还给出了概率论表述，并将集中性结论推广至该情形。此外，我们对先前工作进行了若干简化和改进，特别是在优化问题对偶化、关于ℓ∞距离的集中性，以及与广义KL散度的关联方面。部分结果适用于一般紧凸集，而不仅限于多面体情形。