基于AOC-偏序集的关系概念分析中的收敛问题

中文摘要 AI 辅助

形式概念分析（FCA）是一种从描述一组对象与一组属性的二元表中构建概念分类和发现规则的方法。为处理非二元及更复杂数据，已提出多种扩展方法，如针对多关系数据的关系概念分析（RCA）。RCA旨在突出以与其他对象组的关系为特征的对象组，其底层数据更丰富复杂，因此相比FCA能产出更丰富的结果，但代价是计算和解释复杂度更高。FCA中最常用的概念分类结构是概念格，但在许多应用中，概念格子结构（如AOC-偏序集）更受青睐，要么是为缓解组合爆炸，要么是为聚焦结构中最具信息性的部分。实际上，AOC-偏序集仅表示引入对象或属性的概念，这使其比概念格更小、更易于计算和使用。尽管RCA最初定义在概念格上，但也可实例化于AOC-偏序集。RCA是迭代的，在基于格的设定中其收敛性有保证，但使用AOC-偏序集时该保证失效。本文详细研究这种收敛性的丧失，说明一般情况下收敛性不再有保证的原因，确定仍能保证收敛的条件，讨论如何转换数据集以恢复收敛性，还提出了保留AOC-偏序集结构的收敛变体：关系属性一旦创建就不再移除，以最终结构中可能不存在的概念为代价保证收敛。

英文摘要

Formal Concept Analysis (FCA) is an approach for conceptual classification building and rule discovery from a binary table describing a set of objects by a set of attributes. Extensions have been proposed to deal with non-binary and more complex data, such as Relational Concept Analysis (RCA) for multi-relational data. RCA aims to highlight groups of objects characterized by their relationships with other groups of objects. The richer and more complex nature of the underlying data allows RCA to produce richer results than FCA, at the expense of higher computational and interpretive complexity. The most commonly used conceptual classification structure in FCA is the concept lattice. However, in many applications, concept lattice substructures, such as AOC-posets, are preferred over the full lattice, either to mitigate combinatorial blow-up or to focus on the most informative parts of the structure. Indeed, in an AOC-poset, only concepts introducing an object or an attribute are represented, which makes AOC-posets smaller and easier to compute and use than concept lattices. Although RCA was originally defined on concept lattices, it can also be instantiated on AOC-posets. RCA is iterative and its convergence is guaranteed in the lattice-based setting, but this guarantee is lost when using AOC-posets. In this paper, we investigate this loss of convergence in detail. We show why convergence is no longer guaranteed in the general case, identify conditions under which it can still be ensured, and discuss how a dataset can be transformed to recover convergence. We also propose a convergent variant of the process, which preserves the AOC-poset structure: relational attributes, once created, are never removed, which guarantees convergence at the price of attributes that may refer to concepts absent from the final structures.

发表机构

• Univ. de Strasbourg（斯特拉斯堡大学）
• ENGEES（国立高等环境、农业与工程学校）
• CNRS（法国国家科学研究中心）
• ICube（立方体实验室）
• LIRMM（蒙彼利埃计算机科学、机器人学与微电子实验室）
• Univ. Montpellier（蒙彼利埃大学）

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