# Likelihood Ratio Tests by Kernel Gaussian Embedding

arXiv: https://arxiv.org/abs/2508.07982 · HTML: https://mlsanity.com/paper/2508.07982/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 11 Aug 2025.

## Builds on (most cited first, 15 of 15)

- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [A new graph-based two-sample test for multivariate and object data](https://mlsanity.com/paper/1307.6294/) (2013)
- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [Boosting the Power of Kernel Two-Sample Tests](https://mlsanity.com/paper/2302.10687/) (2023)
- [Kullback-Leibler and Renyi divergences in reproducing kernel Hilbert space and Gaussian process settings](https://mlsanity.com/paper/2207.08406/) (2022)
- [Transportation-Based Functional ANOVA and PCA for Covariance Operators](https://mlsanity.com/paper/2212.04797/) (2022)
- [The Functional Graphical Lasso](https://mlsanity.com/paper/2306.02347/) (2023)
- [Kernel Embeddings and the Separation of Measure Phenomenon](https://mlsanity.com/paper/2505.04613/) (2025)

## Built on by (most cited first, 1 of 1)

- [Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings](https://mlsanity.com/paper/2602.18364/) (2026)

## Abstract (arXiv)

We propose a novel kernel-based nonparametric two-sample test, employing the combined use of kernel mean and kernel covariance embedding. Our test builds on recent results showing how such combined embeddings map distinct probability measures to mutually singular Gaussian measures on the kernel's RKHS. Leveraging this ``separation of measure phenomenon", we construct a test statistic based on the relative entropy between the Gaussian embeddings, in effect the likelihood ratio. The likelihood ratio is specifically tailored to detect equality versus singularity of two Gaussians, and satisfies a ``0/∞" law, in that it vanishes under the null and diverges under the alternative. To implement the test in finite samples, we introduce a regularised version, calibrated by way of permutation. We prove consistency, establish uniform power guarantees under mild conditions, and discuss how our framework unifies and extends prior approaches based on spectrally regularized MMD. Empirical results on synthetic and real data demonstrate remarkable gains in power compared to state-of-the-art methods, particularly in high-dimensional and weak-signal regimes.