# Distance-based and RKHS-based Dependence Metrics in High Dimension

arXiv: https://arxiv.org/abs/1902.03291 · HTML: https://mlsanity.com/paper/1902.03291/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Feb 2019.

## Built on by (most cited first, 5 of 5)

- [Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing](https://mlsanity.com/paper/2210.08149/) (2022)
- [Kernel Two-Sample Tests in High Dimension: Interplay Between Moment Discrepancy and Dimension-and-Sample Orders](https://mlsanity.com/paper/2201.00073/) (2021)
- [Metric Graph Kernels via the Tropical Torelli Map](https://mlsanity.com/paper/2505.12129/) (2025)
- [Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects](https://mlsanity.com/paper/2607.28981/) (2026)
- [Testing Conditional Distribution Equality Using Generative Models](https://mlsanity.com/paper/2610.20447/) (2026)

## Abstract (arXiv)

In this paper, we study distance covariance, Hilbert-Schmidt covariance (aka Hilbert-Schmidt independence criterion [Gretton et al. (2008)]) and related independence tests under the high dimensional scenario. We show that the sample distance/Hilbert-Schmidt covariance between two random vectors can be approximated by the sum of squared componentwise sample cross-covariances up to an asymptotically constant factor, which indicates that the distance/Hilbert-Schmidt covariance based test can only capture linear dependence in high dimension. As a consequence, the distance correlation based t-test developed by Szekely and Rizzo (2013) for independence is shown to have trivial limiting power when the two random vectors are nonlinearly dependent but component-wisely uncorrelated. This new and surprising phenomenon, which seems to be discovered for the first time, is further confirmed in our simulation study. As a remedy, we propose tests based on an aggregation of marginal sample distance/Hilbert-Schmidt covariances and show their superior power behavior against their joint counterparts in simulations. We further extend the distance correlation based t-test to those based on Hilbert-Schmidt covariance and marginal distance/Hilbert-Schmidt covariance. A novel unified approach is developed to analyze the studentized sample distance/Hilbert-Schmidt covariance as well as the studentized sample marginal distance covariance under both null and alternative hypothesis. Our theoretical and simulation results shed light on the limitation of distance/Hilbert-Schmidt covariance when used jointly in the high dimensional setting and suggest the aggregation of marginal distance/Hilbert-Schmidt covariance as a useful alternative.