High-Dimensional BWDM: A Robust Nonparametric Clustering Validation Index for Large-Scale Data

2025-10-15

AI Summary

Determining the optimal number of clusters in high-dimensional, noisy data remains a fundamental challenge in unsupervised learning. To address this, we propose HD-BWDM, a robust nonparametric evaluation framework. HD-BWDM generalizes the Between-Within Distance Metric (BWDM) criterion to high-dimensional spaces by integrating random projection and PCA-based dimensionality reduction to mitigate the curse of dimensionality, and incorporates cluster trimming and center-point distance metrics to enhance robustness against outliers. Theoretically, it provides embedding consistency guarantees grounded in the Johnson–Lindenstrauss lemma. Extensive experiments on multiple high-dimensional and contaminated benchmark datasets demonstrate that HD-BWDM significantly outperforms conventional centroid-based indices—including Calinski–Harabasz and Silhouette—achieving superior stability, interpretability, and reliability. As a theoretically justified, efficient, and robust stopping criterion, HD-BWDM advances practical unsupervised clustering in challenging real-world scenarios.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Determining the appropriate number of clusters in unsupervised learning is a central problem in statistics and data science. Traditional validity indices such as Calinski-Harabasz, Silhouette, and Davies-Bouldin-depend on centroid-based distances and therefore degrade in high-dimensional or contaminated data. This paper proposes a new robust, nonparametric clustering validation framework, the High-Dimensional Between-Within Distance Median (HD-BWDM), which extends the recently introduced BWDM criterion to high-dimensional spaces. HD-BWDM integrates random projection and principal component analysis to mitigate the curse of dimensionality and applies trimmed clustering and medoid-based distances to ensure robustness against outliers. We derive theoretical results showing consistency and convergence under Johnson-Lindenstrauss embeddings. Extensive simulations demonstrate that HD-BWDM remains stable and interpretable under high-dimensional projections and contamination, providing a robust alternative to traditional centroid-based validation criteria. The proposed method provides a theoretically grounded, computationally efficient stopping rule for nonparametric clustering in modern high-dimensional applications.

Problem

• Determining optimal cluster count in unsupervised learning
• Addressing centroid-based index degradation in high dimensions
• Providing robust validation for contaminated large-scale data

Innovation

• Extends BWDM criterion to high-dimensional clustering validation
• Integrates random projection and PCA for dimensionality reduction
• Uses trimmed clustering and medoid distances for outlier robustness