提出一种新型量子统计距离，兼具MMD与Wasserstein优点。
Kernel Trace Distance: Quantum Statistical Metric between Measures through RKHS Density Operators
- 用核协方差算子的Schatten范数定义新距离
- 比MMD更鲁棒且对超参数不敏感，样本复杂度低
- 适合含噪声数据的贝叶斯推断与粒子流模拟
概率分布间的距离是许多统计机器学习任务的核心，如两样本检验和生成建模。本文提出一种通过核协方差算子的Schatten范数比较测度的新距离。该距离属于积分概率度量，介于最大均值差异（MMD）与Wasserstein距离之间。我们证明它克服了MMD的一些缺陷，具有更强的区分能力且对超参数选择更鲁棒。同时继承核方法优势，可缓解高维下的样本复杂度问题。文中提出一种实用算法，引入差异分布的核矩阵扩展，可能具独立研究价值。在污染环境下的近似贝叶斯计算及粒子流模拟中验证了其优越性。
原文摘要 · Abstract (English)
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.
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