# A Kernel Two-Sample Test for Functional Data

arXiv: https://arxiv.org/abs/2008.11095 · HTML: https://mlsanity.com/paper/2008.11095/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Aug 2020.

## Built on by (most cited first, 28 of 28)

- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)
- [Variational Autoencoding Neural Operators](https://mlsanity.com/paper/2302.10351/) (2023)
- [KSD Aggregated Goodness-of-fit Test](https://mlsanity.com/paper/2202.00824/) (2022)
- [Correspondence-Free Domain Alignment for Unsupervised Cross-Domain Image Retrieval](https://mlsanity.com/paper/2302.06081/) (2023)
- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)
- [Bayesian Neural Networks with Domain Knowledge Priors](https://mlsanity.com/paper/2402.13410/) (2024)
- [Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures](https://mlsanity.com/paper/2206.07588/) (2022)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [Energy-Based Models for Functional Data using Path Measure Tilting](https://mlsanity.com/paper/2202.01929/) (2022)
- [A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing](https://mlsanity.com/paper/2302.05686/) (2023)
- [Probability-Flow ODE in Infinite-Dimensional Function Spaces](https://mlsanity.com/paper/2503.10219/) (2025)
- [A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces](https://mlsanity.com/paper/2206.04552/) (2022)
- [A fast and effective kernel two-sample test for large-scale data](https://mlsanity.com/paper/2110.03118/) (2021)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Domain-knowledge Inspired Pseudo Supervision (DIPS) for Unsupervised Image-to-Image Translation Models to Support Cross-Domain Classification](https://mlsanity.com/paper/2303.10310/) (2023)
- [Variable Selection for Comparing High-dimensional Time-Series Data](https://mlsanity.com/paper/2412.06870/) (2024)
- [Anchor-based Maximum Discrepancy for Relative Similarity Testing](https://mlsanity.com/paper/2510.10477/) (2025)
- [Log-PDE Methods for Rough Signature Kernels](https://mlsanity.com/paper/2404.02926/) (2024)
- [Efficient pooling of predictions via kernel embeddings](https://mlsanity.com/paper/2411.16246/) (2024)
- [The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations](https://mlsanity.com/paper/2505.11622/) (2025)
- [Are Two Datasets Close Enough With Statistical Significance? A Kernel Distributional Closeness Testing Approach](https://mlsanity.com/paper/2507.12843/) (2025)
- [Novelty detection on path space](https://mlsanity.com/paper/2512.03243/) (2025)
- [Gaussian mixture models in Hilbert spaces via kernel methods](https://mlsanity.com/paper/2605.05996/) (2026)
- [Functional dynamic mode decomposition: Learning infinite-dimensional systems from data](https://mlsanity.com/paper/2609.29159/) (2026)
- [Kernel-based External Validity of Experimental Studies](https://mlsanity.com/paper/2610.13625/) (2026)

## Abstract (arXiv)

We propose a nonparametric two-sample test procedure based on Maximum Mean Discrepancy (MMD) for testing the hypothesis that two samples of functions have the same underlying distribution, using kernels defined on function spaces. This construction is motivated by a scaling analysis of the efficiency of MMD-based tests for datasets of increasing dimension. Theoretical properties of kernels on function spaces and their associated MMD are established and employed to ascertain the efficacy of the newly proposed test, as well as to assess the effects of using functional reconstructions based on discretised function samples. The theoretical results are demonstrated over a range of synthetic and real world datasets.