任意格上Jaccard距离的三角不等式研究

中文摘要 AI 辅助

本文给出了Jaccard距离对格和实值函数的推广的新理论结果。我们证明当值函数严格正、单调且模性时，Jaccard距离在任意格上满足三角不等式，有效推广了之前高度依赖分配性的结果。转向相对补分配格（其放宽了布尔代数中全局界的要求），我们证明只要值函数正、单调、超模且对数次模，三角不等式就成立。此外，我们将次模值函数的对称差Jaccard公式适配到截面补分配格。转向必要条件，我们证明标准广义Jaccard距离作为有效度量时，超模性是严格要求。最后，我们将这些结构约束放松的实用价值映射到量子信息论、形式概念分析和机器学习等计算领域，并简要展望了开放数学问题。

英文摘要

This paper presents new theoretical results on generalizing the Jaccard distance for lattices and real valuations. We demonstrate that when the valuation is strictly positive, monotone, and modular, the Jaccard distance satisfies the triangle inequality on arbitrary lattices, effectively generalizing earlier results that depended heavily on distributivity. Moving to relatively complemented distributive lattices (which safely drop the requirement for the global bounds found in Boolean algebras), we prove the triangle inequality holds as long as the valuation is positive, monotone, supermodular, and $\log$-submodular. Additionally, we adapt the symmetric-difference Jaccard formulation for submodular valuations to sectionally complemented distributive lattices. Shifting to necessary conditions, we prove that supermodularity is a strict requirement for the standard generalized Jaccard distance to operate as a valid metric. Finally, we map the practical value of relaxing these structural constraints to computational fields like quantum information theory, formal concept analysis, and machine learning, closing with a brief look at open mathematical problems.

发表机构

• University of Craiova（克拉约瓦大学）

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