Sparse $K$-spatial-median clustering for high-dimensional data

2026-05-01

AI Summary

High-dimensional data are often compromised by heavy-tailed distributions, irrelevant features, and complex inter-feature dependencies, which undermine the robustness of conventional clustering methods. This work proposes a robust clustering framework grounded in the spatial median: it replaces the mean with the spatial median in the update step and employs either Euclidean distance or a robust Mahalanobis distance based on the spatial sign covariance matrix during cluster assignment. To address ultra-high dimensionality, the method incorporates a hard feature elimination mechanism that automatically selects a sparsity threshold via the Gap statistic. By jointly integrating robust center estimation, dependence structure modeling, and feature screening, the proposed approach demonstrates substantially improved clustering accuracy and stability over K-means and its sparse variants in simulation studies.

Abstract

We propose a robust clustering framework for high-dimensional data with heavy tails and a large fraction of irrelevant variables. The method replaces the mean updates of Lloyd's $K$-means with \emph{spatial medians} to enhance robustness. For the assignment step, it admits either a Euclidean rule for computational simplicity or a robust Mahalanobis-type metric constructed from the spatial sign covariance matrix to account for heterogeneous scales and feature dependence. To handle the $p \gg n$ regime, we further introduce a simple \emph{hard feature-exclusion} mechanism that removes weakly separating dimensions based on across-center dispersion, with the exclusion threshold selected automatically via a permutation-based Gap criterion. Simulation studies under correlated Gaussian and multivariate $t$ models demonstrate that the proposed approach provides competitive clustering accuracy and improved stability relative to $K$-means and sparse $K$-means baselines.

Problem

• high-dimensional data
• heavy tails
• irrelevant variables
• clustering
• robustness

Innovation

• spatial median
• robust clustering
• feature selection
• high-dimensional data
• sparse K-means