符号矩阵列表可复现性的对数乘积界被证明

该结果解决了 Blondal、Hatami、Hatami、Lalov 和 Tretiak 提出的对数乘积猜想。上界证明通过在立方体的面上构造嵌套无环集合完成：临界点估计提供高连通区域，加权胞腔附着保持集合在各面上相容，再由立方载体定理将其转化为精确有限覆盖和学习规则。

下界方面，Hanner 矩阵给出同阶的界，适用于方阵以及具有不同对数纵横比的显式矩阵族。论文还给出熵倾斜与双侧 Gibbs 构造、面向结构化矩阵的组合上界，并计算了 Paley 矩阵的指数；

同时存在列表可复现性为 4k+1 而等变指数为 3k 的显式类，说明两者加性间隙无界。

Sharp Logarithmic-Product Bounds for List Replicability of Sign Matrices

List replicability measures how many accurate hypotheses suffice to contain the output of a learning rule with arbitrarily high confidence. We prove that every nonempty class of $m$ binary concepts on $n$ domain points has list replicability at most $805\ln(2m)\ln(2n)$. This resolves the logarithmic-product conjecture of Blondal, Hatami, Hatami, Lalov, and Tretiak. Hanner matrices give lower bounds of the same order, for square matrices and for explicit families with unequal logarithmic aspect ratios. The upper bound is obtained by constructing nested acyclic sets in the faces of a cube. A critical-point estimate supplies highly connected regions, weighted cell attachments keep the sets compatible across faces, and a cubical carrier theorem converts them into accurate finite covers and learning rules. Appending one constant coordinate turns list replicability into an exact geometric quantity, the face width, and graph-orientation classes satisfy exact product formulas. In particular, there are explicit classes with list replicability $4k+1$ and equivariant index $3k$, so the additive gap between the two is unbounded. We also develop entropy-tilting and two-sided Gibbs constructions, give combinatorial upper bounds adapted to structured matrices, compute exact values on classical families, and study the index of Paley matrices.

本条情报归属 AITOP「论文」核心栏目，重点聚焦北美前沿实验室与硅谷科技巨头。在跨区域产业观察坐标中，该动态直接关联「AI 开发者工具链、Coding Agent 与工程化实践」研究主线，反映出当前技术演进与商业落地的最新风向。涉及核心实体与技术议题包括：列表可复现性、Hanner矩阵、学习理论、arXiv，读者可通过下方标签进入对应专题页查阅历史脉络与关联对比。

信源核验说明：本文原始线索采编自「arXiv cs.LG」（@Guangjian Zhang），经 AITOP 多维评分管线结构化提炼。