AI 中文总结
该研究开发了基于相关性的框架评估三角劣度函数变体与距离度量的结构一致性，实验发现部分变体一致性高、部分互补，为相关任务的度量选择提供实用见解。
AI 中文摘要
基于三角的度量（通常称为劣度函数）被广泛用于量化距离矩阵偏离理想几何构型的程度。这些函数的不同公式可捕捉局部非均匀性的不同方面，其行为常受评估所用基础距离度量的影响。在实际场景中，尽管规范的劣度函数在概念上更受青睐，但计算成本、算法约束或数据特定特征等因素常要求采用改进版本——例如近似形式或基于不同距离度量定义的替代形式。这引发了一个核心问题：这些变体在多大程度上保留了原始对应物的结构一致性属性？为解决该问题，我们开发了一种基于相关性的系统框架以评估结构一致性。作为该框架的一个示例实例，我们从一组代表性距离矩阵以及随机生成的三角构型中计算劣度序列，这些序列旨在覆盖不同实际场景下可能出现的变体。随后，我们使用四个相关系数评估这些序列之间的成对相似性。实验结果表明，某些劣度变体表现出显著高的结构一致性，而其他变体则呈现互补的行为模式；此外，距离度量的选择对观测到的趋势有显著影响。这些发现为几何重构、三角剖分及成对距离数据的结构分析等任务中明智选择距离度量和三角劣度函数变体提供了实用见解。
英文摘要
Triangle-based measures, commonly referred to as badness functions, are widely employed to quantify the extent to which a distance matrix deviates from an ideal geometric configuration. Different formulations of these functions may capture distinct facets of local non-uniformity, and their behavior is often influenced by the underlying distance metric chosen for evaluation. In practical settings, although a canonical badness function may be conceptually preferred, factors such as computational cost, algorithmic constraints, or data-specific characteristics frequently necessitate the adoption of modified versions-for instance, approximate forms or alternatives defined under different distance metrics. This gives rise to a central question: to what degree do these variants retain the structural consistency properties of their original counterparts? To address this issue, we develop a systematic correlation-based framework for evaluating structural consistency. As an illustrative instantiation of this framework, we compute badness sequences from a set of representative distance matrices alongside randomly generated triangle configurations, which are designed to cover variants that may arise under diverse practical scenarios. We then assess pairwise similarities among these sequences using four correlation coefficients. The experimental outcomes indicate that certain badness variants exhibit a notably high degree of structural consistency, whereas others reveal complementary behavioral patterns; moreover, the choice of distance metric exerts a considerable influence on the observed trends. These findings offer practical insights for the informed selection of distance metrics and triangle badness function variants in tasks including geometric reconstruction, triangulation, and structural analysis of pairwise distance data.
Comments18 pages, 11 tables. Main text in English; includes computational experiments on biological, astronomical, and materials datasets
