发表机构
University of Tennessee（田纳西大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出了一个基于拓扑数据分析中Mapper结构化表示的学习框架，研究了其数学性质，通过时间序列和图分类数据集实验验证了框架的有效性，为相关学习任务提供了实用工具。
AI 中文摘要
现代机器学习（ML）方法对预测任务非常有效，但许多常用表示将复杂数据简化为固定维度的嵌入，可能会抑制多尺度结构组织。来自拓扑数据分析（TDA）的Mapper算法提供了不同视角，它将数据分解为通过神经构造连接的重叠局部区域，产生同时捕获几何组织、局部统计行为和关系连通性的结构化表示。在本研究中，我们开发了一个用于在Mapper诱导的结构化表示上进行学习的框架。我们没有将Mapper视为产生用于下游学习的图的预处理步骤，而是将完整的Mapper构造本身视为表示的一部分。我们研究这些表示的数学性质，包括重标记下的不变性、Mapper表示空间上的距离函数、多尺度分解的结构复杂性以及表示扰动下面向学习的稳定性。在时间序列和图分类数据集上的实验通过对表示消融、Mapper参数敏感性以及诱导表示空间几何的受控研究验证了所提出的框架。这些结果共同证明了所提出的数学框架如何支持对Mapper表示的系统比较、解释和分析，为研究学习任务中的表示几何、结构复杂性和学习稳定性提供了实用工具。
英文摘要
Modern machine learning (ML) methods are highly effective for prediction tasks, but many commonly used representations reduce complex data to fixed dimensional embeddings that may suppress multiscale structural organization. The Mapper algorithm from topological data analysis (TDA) provides a different perspective by decomposing data into overlapping local regions connected through a nerve construction, producing a structured representation that captures geometric organization, local statistical behavior, and relational connectivity simultaneously. In this work, we develop a framework for learning over Mapper induced structured representations. Rather than treating Mapper as a preprocessing step that produces a graph for downstream learning, we treat the full Mapper construction as part of the representation itself. We study mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations. Experiments on time series and graph classification datasets validate the proposed framework through controlled studies of representation ablation, Mapper parameter sensitivity, and the geometry of the induced representation space. Together, these results demonstrate how the proposed mathematical framework enables systematic comparison, interpretation, and analysis of Mapper representations, providing practical tools for studying representation geometry, structural complexity, and learning stability in learning tasks.
Comments29 pages, 10 figures, and 7 tables
