# Kernel Two-Sample Tests for Manifold Data

arXiv: https://arxiv.org/abs/2105.03425 · HTML: https://mlsanity.com/paper/2105.03425/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 78 references that are in the map.
- First posted on arXiv 7 May 2021.

## Builds on (most cited first, 24 of 24)

- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Likelihood Ratios for Out-of-Distribution Detection](https://mlsanity.com/paper/1906.02845/) (2019)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Interpretable Distribution Features with Maximum Testing Power](https://mlsanity.com/paper/1605.06796/) (2016)
- [Fast Two-Sample Testing with Analytic Representations of Probability Measures](https://mlsanity.com/paper/1506.04725/) (2015)
- [A Linear-Time Kernel Goodness-of-Fit Test](https://mlsanity.com/paper/1705.07673/) (2017)
- [A Kernel Two-Sample Test for Functional Data](https://mlsanity.com/paper/2008.11095/) (2020)
- [A new graph-based two-sample test for multivariate and object data](https://mlsanity.com/paper/1307.6294/) (2013)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [LDMNet: Low Dimensional Manifold Regularized Neural Networks](https://mlsanity.com/paper/1711.06246/) (2017)
- [On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives](https://mlsanity.com/paper/1909.03302/) (2019)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [Testing Goodness of Fit of Conditional Density Models with Kernels](https://mlsanity.com/paper/2002.10271/) (2020)
- [Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs](https://mlsanity.com/paper/1910.13476/) (2019)
- [Convergence of Graph Laplacian with kNN Self-tuned Kernels](https://mlsanity.com/paper/2011.01479/) (2020)
- [Change Detection with the Kernel Cumulative Sum Algorithm](https://mlsanity.com/paper/1903.01661/) (2019)
- [Two-sample Statistics Based on Anisotropic Kernels](https://mlsanity.com/paper/1709.05006/) (2017)
- [Goodness-of-fit tests on manifolds](https://mlsanity.com/paper/1909.05229/) (2019)

## Built on by (most cited first, 8 of 8)

- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [Variable Selection for Kernel Two-Sample Tests](https://mlsanity.com/paper/2302.07415/) (2023)
- [Maximum Mean Discrepancy on Exponential Windows for Online Change Detection](https://mlsanity.com/paper/2205.12706/) (2022)
- [A Unified Data Representation Learning for Non-parametric Two-sample Testing](https://mlsanity.com/paper/2412.00613/) (2024)
- [A Manifold Two-Sample Test Study: Integral Probability Metric with Neural Networks](https://mlsanity.com/paper/2205.02043/) (2022)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [On the Intrinsic Dimensions of Data in Kernel Learning](https://mlsanity.com/paper/2601.16139/) (2026)
- [Kernel-based External Validity of Experimental Studies](https://mlsanity.com/paper/2610.13625/) (2026)

## Abstract (arXiv)

We present a study of a kernel-based two-sample test statistic related to the Maximum Mean Discrepancy (MMD) in the manifold data setting, assuming that high-dimensional observations are close to a low-dimensional manifold. We characterize the test level and power in relation to the kernel bandwidth, the number of samples, and the intrinsic dimensionality of the manifold. Specifically, when data densities p and q are supported on a d-dimensional sub-manifold M embedded in an m-dimensional space and are H"older with order β (up to 2) on M, we prove a guarantee of the test power for finite sample size n that exceeds a threshold depending on d, β, and Delta2 the squared L2-divergence between p and q on the manifold, and with a properly chosen kernel bandwidth γ. For small density departures, we show that with large n they can be detected by the kernel test when Delta2 is greater than n^-  2 β/( d + 4 β )  up to a certain constant and γ scales as n-1/(d+4β). The analysis extends to cases where the manifold has a boundary and the data samples contain high-dimensional additive noise. Our results indicate that the kernel two-sample test has no curse-of-dimensionality when the data lie on or near a low-dimensional manifold. We validate our theory and the properties of the kernel test for manifold data through a series of numerical experiments.