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

arXiv: https://arxiv.org/abs/2507.06055 · HTML: https://mlsanity.com/paper/2507.06055/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 8 Jul 2025.

## Builds on (most cited first, 18 of 18)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Approximate Bayesian computation with the Wasserstein distance](https://mlsanity.com/paper/1905.03747/) (2019)
- [McGan: Mean and Covariance Feature Matching GAN](https://mlsanity.com/paper/1702.08398/) (2017)
- [Integral Probability Metrics PAC-Bayes Bounds](https://mlsanity.com/paper/2207.00614/) (2022)
- [(De)-regularized Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/2409.14980/) (2024)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [(f,Gamma)-Divergences: Interpolating between f-Divergences and Integral Probability Metrics](https://mlsanity.com/paper/2011.05953/) (2020)
- [Statistical Optimality and Computational Efficiency of Nystr"om Kernel PCA](https://mlsanity.com/paper/2105.08875/) (2021)
- [On the Sample Complexity of Subspace Learning](https://mlsanity.com/paper/1408.5032/) (2014)
- [Spectral Regularized Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2308.04561/) (2023)
- [Robust and Efficient Approximate Bayesian Computation: A Minimum Distance Approach](https://mlsanity.com/paper/2006.14126/) (2020)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)

## Abstract (arXiv)

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.