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A Unified View of Optimal Kernel Hypothesis Testing

Antonin Schrab

TL;DR

This work presents a unified, minimax-centered view of kernel-based hypothesis testing across three core problems: two-sample testing (MMD), independence testing (HSIC), and goodness-of-fit testing (KSD). It develops adaptive kernel strategies (aggregation and kernel pooling) to achieve power without knowing the optimal kernel, and analyzes uniform separation rates under standard, efficient, private, and robust settings. The results establish minimax-optimal rates for both kernel-based and $L^2$-based alternatives, while detailing practical constructions via permutations and wild bootstrap for non-asymptotic level control. It also highlights open problems, notably in differential privacy lower bounds for $L^2$ separation and in private/robust KSD-GoF testing. Collectively, the framework provides a cohesive blueprint for kernel-based tests with rigorous power guarantees and practical adaptation under realistic constraints, enabling reliable nonparametric hypothesis testing in large-scale settings.

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 across the three frameworks, and open problems are highlighted.

A Unified View of Optimal Kernel Hypothesis Testing

TL;DR

This work presents a unified, minimax-centered view of kernel-based hypothesis testing across three core problems: two-sample testing (MMD), independence testing (HSIC), and goodness-of-fit testing (KSD). It develops adaptive kernel strategies (aggregation and kernel pooling) to achieve power without knowing the optimal kernel, and analyzes uniform separation rates under standard, efficient, private, and robust settings. The results establish minimax-optimal rates for both kernel-based and -based alternatives, while detailing practical constructions via permutations and wild bootstrap for non-asymptotic level control. It also highlights open problems, notably in differential privacy lower bounds for separation and in private/robust KSD-GoF testing. Collectively, the framework provides a cohesive blueprint for kernel-based tests with rigorous power guarantees and practical adaptation under realistic constraints, enabling reliable nonparametric hypothesis testing in large-scale settings.

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 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 across the three frameworks, and open problems are highlighted.

Paper Structure

This paper contains 121 sections, 162 equations, 1 figure.

Table of Contents

  1. Hypothesis testing
  2. Definition of hypothesis testing
  3. Statistical hypothesis testing.
  4. Examples.
  5. Hypothesis test.
  6. Hypothesis tests using discrepancies.
  7. Level of hypothesis testing
  8. Level: type I error.
  9. Test construction (quantile and p-value).
  10. Exchangeability & permutations.
  11. Wild bootstrap.
  12. Computational complexity: permutations and wild bootstrap.
  13. Permutations vs wild bootstrap.
  14. Power of hypothesis testing
  15. Type II error.
  16. ...and 106 more sections

Figures (1)

  • Figure 1: Uniform separation rates in the sample size $N$. Efficiency: block size $B$ ranging from $1$ to $N$, and design size $D$ typically ranging from $N$ to $N^2$. Differential privacy:$(\varepsilon,\delta)$-DP with $\xi = \varepsilon + \log\!({1}/({1-\delta}))$. Robust to data corruption: robust to the corruption of up to $r$ samples. Sobolev regularity: smoothness $s$ and dimension $d$. All uniform separation rates presented hold with logarithmic dependencies in the test errors.