# Testing mutual independence in high dimension via distance covariance

arXiv: https://arxiv.org/abs/1609.09380 · HTML: https://mlsanity.com/paper/1609.09380/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2016.

## Built on by (most cited first, 4 of 4)

- [Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing](https://mlsanity.com/paper/2210.08149/) (2022)
- [Private independence testing across two parties](https://mlsanity.com/paper/2207.03652/) (2022)
- [Causal Learning for Heterogeneous Subgroups Based on Nonlinear Causal Kernel Clustering](https://mlsanity.com/paper/2501.11622/) (2025)
- [On computing and the complexity of computing higher-order U-statistics, exactly](https://mlsanity.com/paper/2508.12627/) (2025)

## Abstract (arXiv)

In this paper, we introduce a mathcal L2 type test for testing mutual independence and banded dependence structure for high dimensional data. The test is constructed based on the pairwise distance covariance and it accounts for the non-linear and non-monotone dependences among the data, which cannot be fully captured by the existing tests based on either Pearson correlation or rank correlation. Our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal. It exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit dimension is high, and is shown to successfully identify nonlinear dependence in empirical data analysis. On the theory side, asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size. As a demonstration of good power properties for our distance covariance based test, we further show that an infeasible version of our test statistic has the rate optimality in the class of Gaussian distribution with equal correlation.