# Adaptivity and Computation-Statistics Tradeoffs for Kernel and Distance based High Dimensional Two Sample Testing

arXiv: https://arxiv.org/abs/1508.00655 · HTML: https://mlsanity.com/paper/1508.00655/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2015.

## Built on by (most cited first, 14 of 14)

- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [Meta Two-Sample Testing: Learning Kernels for Testing with Limited Data](https://mlsanity.com/paper/2106.07636/) (2021)
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- [Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection](https://mlsanity.com/paper/2207.11208/) (2022)
- [A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations](https://mlsanity.com/paper/2502.13570/) (2025)
- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)
- [Kernel Two-Sample Tests in High Dimension: Interplay Between Moment Discrepancy and Dimension-and-Sample Orders](https://mlsanity.com/paper/2201.00073/) (2021)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [A fast and effective kernel two-sample test for large-scale data](https://mlsanity.com/paper/2110.03118/) (2021)
- [Learning Representations for Independence Testing](https://mlsanity.com/paper/2409.06890/) (2024)
- [Nonlinear Sparse Generalized Canonical Correlation Analysis for Multi-view High-dimensional Data](https://mlsanity.com/paper/2502.18756/) (2025)
- [KAIROS: Scalable Model-Agnostic Data Valuation](https://mlsanity.com/paper/2506.23799/) (2025)
- [Adaptive generative moment matching networks for improved learning of dependence structures](https://mlsanity.com/paper/2508.21531/) (2025)
- [Machine-Learning-Assisted Comparison of Regression Functions](https://mlsanity.com/paper/2510.24714/) (2025)

## Abstract (arXiv)

Nonparametric two sample testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. We refer to the most common settings as mean difference alternatives (MDA), for testing differences only in first moments, and general difference alternatives (GDA), which is about testing for any difference in distributions. A large number of test statistics have been proposed for both these settings. This paper connects three classes of statistics - high dimensional variants of Hotelling's t-test, statistics based on Reproducing Kernel Hilbert Spaces, and energy statistics based on pairwise distances. We ask the question: how much statistical power do popular kernel and distance based tests for GDA have when the unknown distributions differ in their means, compared to specialized tests for MDA? We formally characterize the power of popular tests for GDA like the Maximum Mean Discrepancy with the Gaussian kernel (gMMD) and bandwidth-dependent variants of the Energy Distance with the Euclidean norm (eED) in the high-dimensional MDA regime. Some practically important properties include (a) eED and gMMD have asymptotically equal power; furthermore they enjoy a free lunch because, while they are additionally consistent for GDA, they also have the same power as specialized high-dimensional t-test variants for MDA. All these tests are asymptotically optimal (including matching constants) under MDA for spherical covariances, according to simple lower bounds, (b) The power of gMMD is independent of the kernel bandwidth, as long as it is larger than the choice made by the median heuristic, (c) There is a clear and smooth computation-statistics tradeoff for linear-time, subquadratic-time and quadratic-time versions of these tests, with more computation resulting in higher power.