Statistics Seminar: Victor Panaretos (EPFL)

We show that kernel covariance embeddings lead to information-theoretically perfect separation of distinct probability distributions. In statistical terms, we establish that testing for the equality of two probability measures on a compact and separable metric space is equivalent to testing for the singularity between two centred Gaussian measures on a Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure. This “separation of measure phenomenon” appears to be a blessing of infinite dimensionality, and to underpin the outstanding empirical effectiveness of the so-called “kernel trick”. Interestingly, we will show that existing methods will generally fail to harness this phenomenon. By contrast, we will show that the “right” quantity to target is the likelihood ratio of the Gaussian embeddings, which vanishes under the null and diverges under the alternative. And we further show that this phenomenon can be operationalised for the purpose of inference: a regularised version of the likelihood ratio provides a nonparametric test that maintains the level while yielding remarkable gains in power compared to state-of-the-art methods, particularly in high-dimensional and weak-signal regimes. Based on joint work with Leonardo Santoro (EPFL) and Kartik Waghmare (ETH Zürich).