发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出首个多项式时间近似计数与几乎均匀采样两个拟阵公共基的算法，通过软交集松弛和最大熵凸规划实现，并推广至更广场景。
AI 中文摘要
我们设计了首个多项式时间算法，用于近似计数和几乎均匀采样由独立预言机给出的两个拟阵的公共基。此外，我们的算法远远推广到布尔立方体上满足简单非负曲率条件的两个概率测度的哈达玛积。这些算法原语在统计物理、多面体组合学、量子多体系统研究等领域有众多应用。我们的方法有两个关键成分。● 我们通过在对两个输入测度形成的乘积测度上施加重叠惩罚来松弛交集。我们通过一种集成的 Bochner 型方法证明，这个“软交集”在所有外部场上均匀满足 Poincare 不等式。● 我们求解一个对偶最大熵凸规划，以计算在软交集测度下硬约束以高概率满足的外部场。我们直接使用均匀 Poincare 不等式和梯度范数的小性来界定该成功概率。AI 披露：GPT-5.6 Sol Ultra 和 GPT-6 Astra Ultra 被大量用于开发本文的思想。更完整的讨论包含在致谢中。
英文摘要
We design the first polynomial-time algorithms for approximately counting and almost uniformly sampling common bases of two matroids given by their independence oracles. Moreover, our algorithms generalize far beyond this to Hadamard products of two probability measures on the Boolean cube satisfying a simple nonnegative curvature condition. These algorithmic primitives have myriad applications in statistical physics, polyhedral combinatorics, the study of quantum many-body systems, and beyond. Our approach has two key ingredients. $\bullet$ We relax the intersection by imposing an overlap penalty on the product measure formed by the two input measures. We prove, via an integrated Bochner-type method, that this "$\textit{soft intersection}$" satisfies a Poincare inequality uniformly over all external fields. $\bullet$ We solve a dual maximum entropy convex program to compute external fields under which the hard constraint is satisfied with high probability under the soft intersection measure. We bound this success probability directly using the uniform Poincare inequality and smallness of the gradient norm. $\textbf{AI Disclosure}$ GPT-5.6 Sol Ultra and GPT-6 Astra Ultra were heavily used to develop the ideas in this paper. A more complete discussion is included in the acknowledgments.
