Spatial-Sign based High dimensional Change Point Inference

2025-04-27

AI Summary

This paper addresses the challenges of robustness and adaptivity in high-dimensional change-point detection under heavy-tailed distributions and sparse/dense alternatives. We propose a novel generalized CUSUM testing framework that, for the first time, incorporates spatial medians and spatial signs into high-dimensional change-point inference—yielding statistics robust to heavy tails and adaptive across dimensions. By integrating both $L_infty$ (maximum-type) and $L_2$ (sum-type) cross-dimensional aggregation strategies, the method achieves adaptive detection across diverse signal structures. Theoretically, we derive the exact asymptotic null distribution of the test statistic and establish its asymptotic independence. Empirically, the proposed method demonstrates significantly higher detection power than state-of-the-art benchmarks under heavy-tailed noise and across a spectrum of sparsity levels—from highly sparse to dense alternatives.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

High-dimensional changepoint inference, adaptable to diverse alternative scenarios, has attracted significant attention in recent years. In this paper, we propose an adaptive and robust approach to changepoint testing. Specifically, by generalizing the classical mean-based cumulative sum (CUSUM) statistic, we construct CUSUM statistics based on spatial medians and spatial signs. We introduce test statistics that consider the maximum and summation of the CUSUM statistics across different dimensions, respectively, and take the maximum across all potential changepoint locations. The asymptotic distributions of test statistics under the null hypothesis are derived. Furthermore, the test statistics exhibit asymptotic independence under mild conditions. Building on these results, we propose an adaptive testing procedure that combines the max-$L_infty$-type and max-$L_2$-type statistics to achieve high power under both sparse and dense alternatives. Through numerical experiments and theoretical analysis, the proposed method demonstrates strong performance and exhibits robustness across a wide range of signal sparsity levels and heavy-tailed distributions.

Problem

• Proposes adaptive high-dimensional changepoint testing method
• Generalizes CUSUM statistics using spatial medians and signs
• Combines max-L∞ and max-L2 statistics for robustness

Innovation

• Spatial median and sign based CUSUM statistics
• Max and sum CUSUM statistics combination
• Adaptive testing for sparse and dense alternatives