Testing for Homogeneity with Kernel Fisher Discriminant Analysis

We propose to investigate test statistics for testing homogeneity in reproducing kernel Hilbert spaces. Asymptotic null distributions under null hypothesis are derived, and consistency against fixed and local alternatives is assessed. Finally, experimental evidence of the performance of the proposed approach on both artificial data and a speaker verification task is provided.

💡 Research Summary

The paper introduces a novel two‑sample homogeneity test that operates in a reproducing kernel Hilbert space (RKHS) by leveraging Kernel Fisher Discriminant Analysis (KFDA). Traditional kernel‑based two‑sample tests such as Maximum Mean Discrepancy (MMD) or Hilbert‑Schmidt Independence Criterion (HSIC) rely primarily on differences between kernel mean embeddings and do not explicitly exploit within‑class covariance structure. The authors propose to fill this gap by constructing a test statistic from the KFDA direction that maximally separates the two samples while controlling intra‑sample variability.

Methodology
Given independent samples (X={x_i}{i=1}^{n}) and (Y={y_j}{j=1}^{m}) and a positive‑definite kernel (k), each observation is mapped to a feature (\phi(x)) in the RKHS (\mathcal{H}). The empirical means (\mu_X) and (\mu_Y) and the pooled covariance operator (\Sigma = \frac{n}{n+m}\Sigma_X + \frac{m}{n+m}\Sigma_Y) are formed. A regularization parameter (\lambda>0) stabilizes the inverse of (\Sigma). The KFDA direction is defined as

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