Ball Impurity: Measuring Heterogeneity in General Metric Spaces

• 成果类型: Article; Early Access
• 署名作者: Che, Menglu; Li, Ting; Pan, Wenliang; Wang, Xueqin; Zhang, Heping
• 署名单位: Yale University; Southern University of Science & Technology; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; University of Science & Technology of China, CAS
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.2025.2595733
• 发表日期: 2026-03-06
• 关键词: Decision tree; Heterogeneity; Non-Euclidean; community detection; distance; networks

摘要

Data in various domains, such as neuroimaging and network data analysis, often come in complex forms without possessing a Hilbert structure. The complexity necessitates innovative approaches for effective analysis. We propose a novel measure of heterogeneity, ball impurity, which is designed to work with complex non-Euclidean objects. Our approach extends the notion of impurity to general metric spaces, providing a versatile tool for feature selection and tree models. The ball impurity measure exhibits desirable properties, such as the triangular inequality, and is computationally tractable, enhancing its practicality and usefulness. Extensive experiments on synthetic data and real data from the UK Biobank validate the efficacy of our approach in capturing data heterogeneity. Remarkably, our results compare favorably with state-of-the-art methods in metric spaces, highlighting the potential of ball impurity as a valuable tool for addressing complex data analysis tasks. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.