# A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing

arXiv: https://arxiv.org/abs/2302.05686 · HTML: https://mlsanity.com/paper/2302.05686/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 39 references that are in the map.
- First posted on arXiv 11 Feb 2023.

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

- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [A Kernel Two-Sample Test for Functional Data](https://mlsanity.com/paper/2008.11095/) (2020)
- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [Dimension-agnostic inference using cross U-statistics](https://mlsanity.com/paper/2011.05068/) (2020)
- [Asymptotics of the Empirical Bootstrap Method Beyond Asymptotic Normality](https://mlsanity.com/paper/2011.11248/) (2020)

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

- [On the Robustness of Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2408.05854/) (2024)
- [Kernel Tests of Equivalence](https://mlsanity.com/paper/2603.10886/) (2026)
- [Diagonal Symmetrization of Neural Network Solvers for the Many-Electron Schr"odinger Equation](https://mlsanity.com/paper/2502.05318/) (2025)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [On Ranking-based Tests of Independence](https://mlsanity.com/paper/2403.07464/) (2024)

## Abstract (arXiv)

We prove a convergence theorem for U-statistics of degree two, where the data dimension d is allowed to scale with sample size n. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite n and d, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with d and the bandwidth.