A Martingale Kernel Independence Test

TL;DR

This paper introduces two new martingale-based independence tests, $m\mathrm{HSIC}$ and $md\mathrm{HSIC}$, that are faster and maintain accuracy compared to permutation-based methods, by using standard normal null distributions.

Contribution

The authors adapt martingale MMD construction to independence testing, creating tests with normal null distributions that eliminate permutation calibration, improving speed and scalability.

Findings

Both tests match permutation-based baselines in error rate and power.

They run 25 to 60 times faster than permutation methods.

The tests are effective for high-dimensional data with up to 500 features.

Abstract

The Hilbert-Schmidt Independence Criterion (HSIC) and its joint-independence extension $d\mathrm{HSIC}$ are degenerate $V$-statistics whose data-dependent weighted-$\chi^2$ null limits force a permutation calibration that multiplies the per-test cost by the number of permutations, in practice two orders of magnitude. Adapting the recent martingale MMD construction for two-sample testing to the (joint) independence problem, we introduce two studentised statistics whose null distributions are standard normal regardless of the data law, so that a single normal-quantile lookup replaces the permutation step entirely. The first, $m\mathrm{HSIC}$, is a self-normalised lower-triangular sum of the Hadamard product of two empirically centred Gram matrices. Under independence and bounded-fourth-moment kernels it converges to a standard normal. It is consistent against every fixed alternative, and runs at quadratic cost in the sample size without any sample split, matching the biased HSIC $V$-statistic. Our second statistic, $md\mathrm{HSIC}$, achieves finite-sample consistency with a single half-sample split: the centring is estimated on one half and the lower-triangular self-normalised martingale is run on the other, shrinking the conditional-mean residual to a quantity that is exponentially small in $d$, so the statistic is asymptotically standard normal at every fixed number of jointly tested variables, with a per-test cost that grows only linearly in $d$. On synthetic data with per-variable input dimension from $1$ to $500$ and between $2$ and $10$ jointly tested variables, both statistics match the empirical type-I error rate and test power of permutation-calibrated baselines while running $25$ to $60\times$ faster.