# The Minimax Rate of HSIC Estimation for Translation-Invariant Kernels

arXiv: https://arxiv.org/abs/2403.07735 · HTML: https://mlsanity.com/paper/2403.07735/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 63 references that are in the map.
- First posted on arXiv 12 Mar 2024.

## Builds on (most cited first, 13 of 13)

- [Characteristic and Universal Tensor Product Kernels](https://mlsanity.com/paper/1708.08157/) (2017)
- [Kernel-based Tests for Joint Independence](https://mlsanity.com/paper/1603.00285/) (2016)
- [Distinguishing cause from effect using observational data: methods and benchmarks](https://mlsanity.com/paper/1412.3773/) (2014)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Global Sensitivity Analysis with Dependence Measures](https://mlsanity.com/paper/1311.2483/) (2013)
- [A Permutation-Free Kernel Independence Test](https://mlsanity.com/paper/2212.09108/) (2022)
- [A Kernel Test for Three-Variable Interactions](https://mlsanity.com/paper/1306.2281/) (2013)
- [Distance Metrics for Measuring Joint Dependence with Application to Causal Inference](https://mlsanity.com/paper/1711.09179/) (2017)
- [On Distance and Kernel Measures of Conditional Independence](https://mlsanity.com/paper/1912.01103/) (2019)
- [Nystr"om M-Hilbert-Schmidt Independence Criterion](https://mlsanity.com/paper/2302.09930/) (2023)
- [Nonparametric Independence Testing for Small Sample Sizes](https://mlsanity.com/paper/1406.1922/) (2014)
- [Optimal Uncertainty Quantification of a risk measurement from a thermal-hydraulic code using Canonical Moments](https://mlsanity.com/paper/1901.07903/) (2019)
- [Wind power predictions from nowcasts to 4-hour forecasts: a learning approach with variable selection](https://mlsanity.com/paper/2204.09362/) (2022)

## Built on by (most cited first, 2 of 2)

- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)

## Abstract (arXiv)

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≥ 2 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 Rd for Borel measures containing the Gaussians with continuous bounded translation-invariant characteristic kernels is mathcal O!left(n-1/2right). 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 Nystr"om-based one) on mathbb Rd.