# Learning Deep Kernels for Non-Parametric Two-Sample Tests

arXiv: https://arxiv.org/abs/2002.09116 · HTML: https://mlsanity.com/paper/2002.09116/

## Key facts

- Cited by 130 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2020.

## Built on by (most cited first, 30 of 130)

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- [Federated Class-Incremental Learning](https://mlsanity.com/paper/2203.11473/) (2022)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
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- [Beyond Reward Hacking: Causal Rewards for Large Language Model Alignment](https://mlsanity.com/paper/2501.09620/) (2025)
- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Meta Discovery: Learning to Discover Novel Classes given Very Limited Data](https://mlsanity.com/paper/2102.04002/) (2021)
- [Evaluating generative models in high energy physics](https://mlsanity.com/paper/2211.10295/) (2022)
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- [Learning Bounds for Open-Set Learning](https://mlsanity.com/paper/2106.15792/) (2021)
- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [A Witness Two-Sample Test](https://mlsanity.com/paper/2102.05573/) (2021)
- [KSD Aggregated Goodness-of-fit Test](https://mlsanity.com/paper/2202.00824/) (2022)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [AutoML Two-Sample Test](https://mlsanity.com/paper/2206.08843/) (2022)
- [Detecting Adversarial Data by Probing Multiple Perturbations Using Expected Perturbation Score](https://mlsanity.com/paper/2305.16035/) (2023)
- [Detecting Machine-Generated Texts by Multi-Population Aware Optimization for Maximum Mean Discrepancy](https://mlsanity.com/paper/2402.16041/) (2024)
- [A Learning Based Hypothesis Test for Harmful Covariate Shift](https://mlsanity.com/paper/2212.02742/) (2022)
- [Isometric Propagation Network for Generalized Zero-shot Learning](https://mlsanity.com/paper/2102.02038/) (2021)
- [Meta Two-Sample Testing: Learning Kernels for Testing with Limited Data](https://mlsanity.com/paper/2106.07636/) (2021)
- [On the benefits of representation regularization in invariance based domain generalization](https://mlsanity.com/paper/2105.14529/) (2021)
- [TOHAN: A One-step Approach towards Few-shot Hypothesis Adaptation](https://mlsanity.com/paper/2106.06326/) (2021)
- [Understanding Multimodal LLMs Under Distribution Shifts: An Information-Theoretic Approach](https://mlsanity.com/paper/2502.00577/) (2025)
- [Neural Tangent Kernel Maximum Mean Discrepancy](https://mlsanity.com/paper/2106.03227/) (2021)
- [Differentially Private Neural Tangent Kernels for Privacy-Preserving Data Generation](https://mlsanity.com/paper/2303.01687/) (2023)
- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [Model-agnostic out-of-distribution detection using combined statistical tests](https://mlsanity.com/paper/2203.01097/) (2022)
- [DC-Check: A Data-Centric AI checklist to guide the development of reliable machine learning systems](https://mlsanity.com/paper/2211.05764/) (2022)

## Abstract (arXiv)

We propose a class of kernel-based two-sample tests, which aim to determine whether two sets of samples are drawn from the same distribution. Our tests are constructed from kernels parameterized by deep neural nets, trained to maximize test power. These tests adapt to variations in distribution smoothness and shape over space, and are especially suited to high dimensions and complex data. By contrast, the simpler kernels used in prior kernel testing work are spatially homogeneous, and adaptive only in lengthscale. We explain how this scheme includes popular classifier-based two-sample tests as a special case, but improves on them in general. We provide the first proof of consistency for the proposed adaptation method, which applies both to kernels on deep features and to simpler radial basis kernels or multiple kernel learning. In experiments, we establish the superior performance of our deep kernels in hypothesis testing on benchmark and real-world data. The code of our deep-kernel-based two sample tests is available at https://github.com/fengliu90/DK-for-TST.