# A simpler condition for consistency of a kernel independence test

arXiv: https://arxiv.org/abs/1501.06103 · HTML: https://mlsanity.com/paper/1501.06103/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2015.

## Built on by (most cited first, 10 of 10)

- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [Random Smoothing Regularization in Kernel Gradient Descent Learning](https://mlsanity.com/paper/2305.03531/) (2023)
- [Bandits for Learning to Explain from Explanations](https://mlsanity.com/paper/2102.03815/) (2021)
- [Discussion of `Multiscale Fisher's Independence Test for Multivariate Dependence'](https://mlsanity.com/paper/2206.11142/) (2022)
- [Mixture Proportion Estimation and Weakly-supervised Kernel Test for Conditional Independence](https://mlsanity.com/paper/2604.07191/) (2026)
- [Semiparametrically Efficient Inference for Kernel Measures of Noise Heterogeneity](https://mlsanity.com/paper/2605.27526/) (2026)
- [Aggregation of Statistical Evidence under Exchangeability](https://mlsanity.com/paper/2607.15823/) (2026)

## Abstract (arXiv)

A statistical test of independence may be constructed using the Hilbert-Schmidt Independence Criterion (HSIC) as a test statistic. The HSIC is defined as the distance between the embedding of the joint distribution, and the embedding of the product of the marginals, in a Reproducing Kernel Hilbert Space (RKHS). It has previously been shown that when the kernel used in defining the joint embedding is characteristic (that is, the embedding of the joint distribution to the feature space is injective), then the HSIC-based test is consistent. In particular, it is sufficient for the product of kernels on the individual domains to be characteristic on the joint domain. In this note, it is established via a result of Lyons (2013) that HSIC-based independence tests are consistent when kernels on the marginals are characteristic on their respective domains, even when the product of kernels is not characteristic on the joint domain.