AI 中文总结
该研究刻画了凸概率测度集的极值点，关联φ-散度与绝对连续性极小性，将结果应用于多类测度集场景以补充文献结论。
AI 中文摘要
本研究针对任意可测空间(Ω,ℱ)上所有概率测度构成的集合𝒫，以及该空间上有限变差的符号测度构成的仿射集H，证明了凸概率测度集ℳ=𝒫∩H的极值点的若干等价刻画。首先，我们给出了“极值测度具有最小支撑集”这一启发式思想的精确测度论表述。随后，我们将其与关于绝对连续性的极小性概念相联系，证明了ℳ中不支配其他任何元素的点，是实现任意合适φ-散度的某一散度映射爆破的唯一点。最后，考虑另一类φ-散度，我们将ℳ的极值点刻画为相对于任意合适支配测度的φ-散度的严格局部极大值点。我们将所得结果应用于有限空间、积分约束定义的测度集、多边际耦合以及支配概率测度集等场景，以恢复并补充文献中的相关结果。
英文摘要
In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form $\mathcal{M}=\mathcal{P} \cap H$, where $\mathcal{P}$ denotes the set of all probability measures on an arbitrary measurable space $(Ω,\mathcal{F})$ and $H$ is an affine set of signed measures on $(Ω,\mathcal{F})$ with finite variation. We first give a precise measure-theoretic formulation of the heuristic that extreme measures have minimal support. We then connect this with the notion of minimality with respect to absolute continuity, and prove that points that dominate no other element of $\mathcal{M}$ are the only ones realizing the blow-up of a certain divergence map for any suitable $ϕ$-divergence. Finally, considering a different class of $ϕ$-divergences, we recover a characterization of the extreme points of $\mathcal{M}$ as strict local maximizers of $ϕ$-divergences relative to any suitable dominating measures. We apply our result to recover and complement results from the literature in the context of finite spaces, sets of measures defined by integral constraints, multi-marginal couplings, and dominated sets of probability measures.
