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 addresses the two‑sample homogeneity problem—testing whether two probability distributions (P_{1}) and (P_{2}) are identical—within the framework of reproducing kernel Hilbert spaces (RKHS). Classical tests such as Kolmogorov‑Smirnov or Cramér‑von‑Mises rely on cumulative distribution functions and perform well only in low‑dimensional settings; they are insensitive to high‑frequency or localized differences that often arise in functional or structured data (e.g., strings, graphs). Recent work (Gretton et al., 2006) recasts the problem in a kernel setting using the Maximum Mean Discrepancy (MMD), which essentially measures an (L_{2}) distance between kernel density estimators. The authors propose a more powerful alternative that explicitly incorporates the covariance structure of the underlying distributions by employing Kernel Fisher Discriminant Analysis (KFDA).

Key definitions: for a bounded, characteristic kernel (k), the associated RKHS (\mathcal{H}) admits a mean element (\mu_{P}) satisfying (\langle \mu_{P},f\rangle = \mathbb{E}_{P}