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A Kernel Two-Sample Test Invariant under Group Action with Applications to Functional Data

Madison Giacofci, Anouar Meynaoui, Alex Podgorny

Abstract

We introduce a kernel-based two-sample test for comparing probability distributions up to group actions. Our construction yields invariant kernels for locally compact $σ$-compact groups and extends classical Haar-based approaches beyond the compact setting. The resulting invariant Maximum Mean Discrepancy (MMD) test is developed in a general framework where the sample space is assumed to be Polish. Under natural conditions, the invariant kernel induces a characteristic kernel on the quotient space, ensuring consistency of the associated MMD test. The method is well suited to functional data, where invariances such as temporal shifts arise naturally, and its effectiveness is illustrated through simulation studies.

A Kernel Two-Sample Test Invariant under Group Action with Applications to Functional Data

Abstract

We introduce a kernel-based two-sample test for comparing probability distributions up to group actions. Our construction yields invariant kernels for locally compact -compact groups and extends classical Haar-based approaches beyond the compact setting. The resulting invariant Maximum Mean Discrepancy (MMD) test is developed in a general framework where the sample space is assumed to be Polish. Under natural conditions, the invariant kernel induces a characteristic kernel on the quotient space, ensuring consistency of the associated MMD test. The method is well suited to functional data, where invariances such as temporal shifts arise naturally, and its effectiveness is illustrated through simulation studies.
Paper Structure (19 sections, 6 theorems, 88 equations, 7 figures)

This paper contains 19 sections, 6 theorems, 88 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Review on Kernel two-sample tests
  4. Group actions
  5. Invariant two-sample tests under group actions
  6. Weighting and admissible measures
  7. Averaged and invariant kernels
  8. Invariant MMD test in practice
  9. Simulations
  10. General setup
  11. Time shifts of periodic signals
  12. Time shifts of aperiodic signals
  13. Real data application
  14. Proofs
  15. Proof of Proposition \ref{['prop:S_nu']}
  16. ...and 4 more sections

Key Result

Proposition 1

Let $\nu\in \mathcal{M}^\rho(G)$. For all probability measures $P \in \mathcal{P}(\mathcal{X})$, the averaged measure is well-defined and belongs to $\mathcal{M}^\rho(\mathcal{X})$.

Figures (7)

  • Figure 1: Periodic case. Empirical rejection rates of the MMD tests based on $k$, $k^\rho_\lambda$ and the align-then-test baseline with respect to $\delta$. The approximation budget of $k^\rho_\lambda$ is $S=16$. Left.$P \neq Q$ and $\Pi_*P = \Pi_*Q$. Right.$\Pi_*P\neq\Pi_*Q$.
  • Figure 2: Periodic case. Empirical rejection rates of the MMD tests based on $k^\rho_\lambda$with two different approximation budgets. Left.$P \neq Q$ and $\Pi_*P = \Pi_*Q$. Right.$\Pi_*P\neq\Pi_*Q$.
  • Figure 3: Aperiodic case. Empirical rejection rates of the MMD tests based on $k$, $k^\rho_\lambda$ and the align-then-test baseline with respect to $\delta$. The approximation budget of $k^\rho_\lambda$ is $S=16$. Left.$P \neq Q$ and $\Pi_*P = \Pi_*Q$. Right.$\Pi_*P\neq\Pi_*Q$.
  • Figure 4: Aperiodic case. Empirical rejection rates of the MMD tests based on $k^\rho_\lambda$with two different approximation budgets. Left.$P \neq Q$ and $\Pi_*P = \Pi_*Q$. Right.$\Pi_*P\neq\Pi_*Q$.
  • Figure 5: Illustration of the two cycle-extraction procedures on a single recording. In the $S_1$-aligned extraction, an interval $[s_1, s_1+\widehat{T})$ is extracted starting from a detected $s_1$ position. In the random extraction, an interval $[t_0, t_0+\widehat{T})$ is extracted from a randomly selected starting time $t_0$. Dashed vertical lines mark the beginning and end of the extracted interval.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 1
  • Theorem 1
  • Remark 2