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arXiv 2404.03082cs.LG
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基于偏序集的机器学习与数据分析:一项综述

Machine Learning and Data Analysis Using Posets: A Survey

Arnauld Mesinga Mwafise

AI总结:

本综述针对偏序集在机器学习与数据分析应用的研究缺口,提出四轴分类体系,梳理相关模型算法与资源,并明确下一代序感知机器学习的研究议程。

摘要:

偏序集(posets)是一种离散数学结构,可形式化对象间的比较概念,且不强制要求每对对象都具备可比性。这使得偏序集成为机器学习与数据分析诸多场景的自然表示,这类场景中对象间的关系为支配、包含、优先级或精化关系,而非单一标量分数。过去二十年间,大量分散的文献将偏序集与格理论应用于排序、聚类、形式概念分析、多维与多准则数据分析、结构化与安全学习、图与拓扑深度学习及可解释人工智能,覆盖广泛应用领域。尽管相关研究活跃,但该领域存在两项不足:一是缺乏将这些分散研究分支关联起来的组织分类体系;二是缺乏涵盖2025-2026年最新进展的综述,这些进展包括偏序结构化强化学习的安全层、偏序值神经池化算子、多参数持续同调的函子微积分方法。本综述针对这两项不足展开研究:提出了基于偏序方法的四轴分类体系(表示、学习范式、数据模态与任务),提供了沿该分类体系组织的代表性模型与算法的全面比较综述,整理了大量数据集、软件包及算法资源,最后批判性讨论了开放的理论与实际问题,包括模型深度与表达能力、可扩展性-保真度权衡、序结构化数据的异质性以及偏序随时间的动态性,这些问题构成了下一代序感知机器学习的研究议程。

英文摘要:

Partially ordered sets (posets) are discrete mathematical structures that formalize the notion of comparison without forcing every pair of objects to be comparable. This makes them a natural representation for the many machine learning and data-analysis settings in which objects are related by dominance, containment, priority, or refinement relations rather than by a single scalar score. Over the past two decades, a substantial and fragmented literature has connected posets and lattice theory to ranking, clustering, formal concept analysis, multidimensional and multi-criteria data analysis, structured and safe learning, graph and topological deep learning, and explainable artificial intelligence, spanning a wide range of application domains. Despite this activity, the field has lacked (i) an organizing taxonomy that relates these disparate strands of work, and (ii) an up-to-date account extending through 2025--2026 that incorporates recent developments in poset-structured learning methods, including poset-structured safety layers for reinforcement learning, poset-valued neural pooling operators, functor-calculus approaches to multiparameter persistent homology. This survey addresses both gaps. We propose a four-axis taxonomy of poset-based methods (representation, learning paradigm, data modality, and task), provide a comprehensive and comparative review of representative models and algorithms organized along this taxonomy, curate an extensive collection of datasets, software packages, and algorithmic resources, and close with a critical discussion of open theoretical and practical problems -- including model depth and expressivity, scalability--fidelity trade-offs, heterogeneity of order-structured data, and dynamicity of posets over time -- that we argue define a research agenda for the next generation of order-aware machine learning.

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