Kernel Trace Distance: Quantum Statistical Metric between Measures through RKHS Density Operators

Distances between probability distributions are a key component of many statistical machine learning tasks, from two-sample testing to generative modeling, among others. We introduce a novel distance between measures that compares them through a Schatten norm of their kernel covariance operators. We show that this new distance is an integral probability metric that can be framed between a Maximum Mean Discrepancy (MMD) and a Wasserstein distance. In particular, we show that it avoids some pitfalls of MMD, by being more discriminative and robust to the choice of hyperparameters. Moreover, it benefits from some compelling properties of kernel methods, that can avoid the curse of dimensionality for their sample complexity. We provide an algorithm to compute the distance in practice by introducing an extension of kernel matrix for difference of distributions that could be of independent interest. Those advantages are illustrated by robust approximate Bayesian computation under contamination as well as particle flow simulations.

Fast kernel half-space depth for data with non-convex supports

Data depth is a statistical function that generalizes order and quantiles to the multivariate setting and beyond, with applications spanning over descriptive and visual statistics, anomaly detection, testing, etc. The celebrated halfspace depth exploits data geometry via an optimization program to deliver properties of invariances, robustness, and non-parametricity. Nevertheless, it implicitly assumes convex data supports and requires exponential computational cost. To tackle distribution's multimodality, we extend the halfspace depth in a Reproducing Kernel Hilbert Space (RKHS). We show that the obtained depth is intuitive and establish its consistency with provable concentration bounds that allow for homogeneity testing. The proposed depth can be computed using manifold gradient making faster than halfspace depth by several orders of magnitude. The performance of our depth is demonstrated through numerical simulations as well as applications such as anomaly detection on real data and homogeneity testing.