@inproceedings{
kalinke2024the,
title={The Minimax Rate of {HSIC} Estimation for Translation-Invariant Kernels},
author={Florian Kalinke and Zolt{\'a}n Szab{\'o}},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=KyNO0n1bJ9}
}
Kernel techniques are among the most influential approaches in data science and statistics. Under mild conditions, the reproducing kernel Hilbert space associated to a kernel is capable of encoding the independence of $M\ge2$ random variables. Probably the most widespread independence measure relying on kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also referred to as distance covariance in the statistics literature). Despite various existing HSIC estimators designed since its introduction close to two decades ago, the fundamental question of the rate at which HSIC can be estimated is still open. In this work, we prove that the minimax optimal rate of HSIC estimation on $\mathbb{R}^d$ for Borel measures containing the Gaussians with continuous bounded translation-invariant  characteristic  kernels  is $\mathcal{O}\left(n^{-1/2}\right)$. Specifically, our result implies the optimality in the minimax sense of many of the most-frequently used estimators (including the U-statistic, the V-statistic, and the Nyström-based one) on $\mathbb{R}^d$.
The Minimax Rate of HSIC Estimation for Translation-Invariant Kernels
Florian Kalinke Zoltán Szabó