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

TL;DR

This paper introduces a new quantum statistical metric called Kernel Trace Distance, which compares measures via RKHS density operators, offering advantages over existing metrics like MMD and Wasserstein in terms of discrimination and robustness.

Contribution

It proposes a novel distance based on Schatten norms of kernel covariance operators, bridging MMD and Wasserstein, with practical algorithms and applications in Bayesian computation and particle flow.

Findings

More discriminative than MMD

Robust to hyperparameter choices

Avoids curse of dimensionality in sample complexity

Abstract

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.