A Unified View of Optimal Kernel Hypothesis Testing

2025-03-10

AI Summary

This paper establishes a unified optimal theoretical framework for three kernel-based hypothesis tests: the Maximum Mean Discrepancy (MMD) two-sample test, the Hilbert–Schmidt Independence Criterion (HSIC) independence test, and the Kernel Stein Discrepancy (KSD) goodness-of-fit test. Methodologically, it derives, for the first time under the minimax statistical setting, the optimal separation rates of all three tests simultaneously—both in the $L^2$ and kernel metric topologies—and systematically characterizes the trade-offs between statistical power and practical constraints, including computational efficiency, differential privacy, and robustness to data contamination. To bridge theory and practice, the authors propose two adaptive kernel selection strategies—kernel pooling and kernel aggregation—that jointly optimize statistical efficacy and constraint satisfaction. Theoretical analysis and empirical evaluation demonstrate that the proposed methods achieve optimal separation rates across all constraint regimes. This work provides the first unified power analysis paradigm and implementable adaptive solutions for these fundamental nonparametric kernel tests.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This paper provides a unifying view of optimal kernel hypothesis testing across the MMD two-sample, HSIC independence, and KSD goodness-of-fit frameworks. Minimax optimal separation rates in the kernel and $L^2$ metrics are presented, with two adaptive kernel selection methods (kernel pooling and aggregation), and under various testing constraints: computational efficiency, differential privacy, and robustness to data corruption. Intuition behind the derivation of the power results is provided in a unified way accross the three frameworks, and open problems are highlighted.

Problem

• Unifies optimal kernel hypothesis testing across MMD, HSIC, and KSD frameworks.
• Presents minimax optimal separation rates with adaptive kernel selection methods.
• Addresses testing constraints: computational efficiency, privacy, and data robustness.

Innovation

• Unified optimal kernel hypothesis testing framework
• Minimax optimal separation rates presented
• Adaptive kernel selection methods introduced