OpenAI 构造首个 χ 与 χ* 不等的自由熵反例，arXiv 论文给出详细分析
On counterexamples to the unification of free entropy
OpenAI 这次不是发模型，是解了一个几十年的数学开放问题，构造出 χ 与 χ* 首次分离的例子。这篇论文把背后的信息几何讲清楚了，数学背景的读者可以细看。
Voiculescu 提出的两种自由熵 χ 与 χ* 何时相等是一个长期悬而未决的问题。OpenAI 最近构造出首个满足 −∞ < χ < χ* 的具体例子，打破了两者的统一预期。这篇 arXiv 论文给出类似例子的详细展示，并通过 Wasserstein 距离证明自由 Fisher 信息与经典 Fisher 信息大 n 极限之间存在分离，把该例子置于概率论与信息几何的框架中解释。
On counterexamples to the unification of free entropy
It has been a long-standing open problem to determine when two versions of Voiculescu's free entropy, $χ$ and $χ^*$, agree. OpenAI recently generated the first example with $-\infty < χ< χ^*$. We give a detailed presentation of a similar example, and explicitly show the separation between free Fisher information and the large-$n$ limit of classical Fisher information through the relationship with Wasserstein distance. We thus situate the example in the context of probability theory and information geometry.
