From Two Sample Testing to Singular Gaussian Discrimination

• URL: http://arxiv.org/abs/2505.04613v1
• Date: Wed, 07 May 2025 17:56:19 GMT
• Title: From Two Sample Testing to Singular Gaussian Discrimination
• Authors: Leonardo V. Santoro, Kartik G. Waghmare, Victor M. Panaretos,
• Abstract summary: Discerning two singular Gaussians is simpler from an information-theoretic perspective than non-parametric two-sample testing.<n>This appears to be a new instance of the blessing of dimensionality that can be harnessed for the design of efficient inference tools in great generality.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We establish that testing for the equality of two probability measures on a general separable and compact metric space is equivalent to testing for the singularity between two corresponding Gaussian measures on a suitable Reproducing Kernel Hilbert Space. The corresponding Gaussians are defined via the notion of kernel mean and covariance embedding of a probability measure. Discerning two singular Gaussians is fundamentally simpler from an information-theoretic perspective than non-parametric two-sample testing, particularly in high-dimensional settings. Our proof leverages the Feldman-Hajek criterion for singularity/equivalence of Gaussians on Hilbert spaces, and shows that discrepancies between distributions are heavily magnified through their corresponding Gaussian embeddings: at a population level, distinct probability measures lead to essentially separated Gaussian embeddings. This appears to be a new instance of the blessing of dimensionality that can be harnessed for the design of efficient inference tools in great generality.

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