# A Martingale Kernel Independence Test

arXiv: https://arxiv.org/abs/2605.22549 · HTML: https://mlsanity.com/paper/2605.22549/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 9 references that are in the map.
- First posted on arXiv 21 May 2026.

## Builds on (most cited first, 6 of 6)

- [Kernel-based Tests for Joint Independence](https://mlsanity.com/paper/1603.00285/) (2016)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (2021)
- [Sequential Nonparametric Testing with the Law of the Iterated Logarithm](https://mlsanity.com/paper/1506.03486/) (2015)
- [A Permutation-Free Kernel Independence Test](https://mlsanity.com/paper/2212.09108/) (2022)
- [Permutation-Free High-Order Interaction Tests](https://mlsanity.com/paper/2506.05963/) (2025)
- [A Martingale Kernel Two-Sample Test](https://mlsanity.com/paper/2510.11853/) (2025)

## Abstract (arXiv)

The Hilbert-Schmidt Independence Criterion (HSIC) and its joint-independence extension dHSIC are degenerate V-statistics whose data-dependent weighted-chi2 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, mHSIC, 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, mdHSIC, 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× faster.