From Separation to Transport: Second-Order Kernel Methods

16.10.2026 11:15 – 12:15

RESEARCH INSTITUTE FOR STATISTICS AND INFORMATION SCIENCE SEMINARS

ABSTRACT

Kernel methods provide a powerful way to represent data distributions. We argue that their full potential can be understood through second-order embeddings, equipped with a geometry appropriate to the task at hand. For distributional inference, an information-theoretic perspective leads to a striking separation-of-measures phenomenon: distinct distributions are mapped to mutually singular Gaussian embeddings in the kernel space. This yields a likelihood-ratio interpretation of kernel two-sample testing and provides a unified view of several classical methods. This representational richness extends beyond testing to generative sampling, which calls for a geometry capturing the transport structure between distributions. We use the Bures–Wasserstein geometry of Gaussian embeddings to construct a neural-network-free framework for generative sampling. The resulting closed-form geodesic induces an affine velocity field in the RKHS, which we pull back to the data space to obtain an explicit transport map. A geometric diagnostic then allows the velocity field to be iteratively refined. Together, these results illustrate how second-order kernel embeddings, combined with task-specific geometries, can provide a common framework for statistical inference and generative modeling.

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