Kernel Two-Sample Testing via Directional Components Analysis

2025-08-11

AI Summary

To address the low statistical power of kernel two-sample tests in high-dimensional and imbalanced-sample settings, this paper proposes a direction-adaptive test based on spectral decomposition of the Maximum Mean Discrepancy (MMD). The method identifies dominant directional components in the Reproducing Kernel Hilbert Space (RKHS) that exhibit stable estimation and high signal-to-noise ratio, via spectral analysis of the MMD statistic; it then constructs a robust discriminative statistic by fusing multi-kernel information. Critical values are efficiently approximated using the multiplier bootstrap. Experiments demonstrate that the proposed method significantly improves statistical power while strictly controlling Type-I error rates—outperforming existing MMD-based baselines—and achieves computational efficiency. Its practical utility is validated on real microarray datasets.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We propose a novel kernel-based two-sample test that leverages the spectral decomposition of the maximum mean discrepancy (MMD) statistic to identify and utilize well-estimated directional components in reproducing kernel Hilbert space (RKHS). Our approach is motivated by the observation that the estimation quality of these components varies significantly, with leading eigen-directions being more reliably estimated in finite samples. By focusing on these directions and aggregating information across multiple kernels, the proposed test achieves higher power and improved robustness, especially in high-dimensional and unbalanced sample settings. We further develop a computationally efficient multiplier bootstrap procedure for approximating critical values, which is theoretically justified and significantly faster than permutation-based alternatives. Extensive simulations and empirical studies on microarray datasets demonstrate that our method maintains the nominal Type I error rate and delivers superior power compared to other existing MMD-based tests.

Problem

• Improves kernel-based two-sample testing via directional components
• Enhances power and robustness in high-dimensional unbalanced settings
• Develops efficient bootstrap for critical values faster than permutations

Innovation

• Spectral decomposition of MMD for directional components
• Aggregates information across multiple kernels
• Efficient multiplier bootstrap for critical values