# Characteristic and Universal Tensor Product Kernels

arXiv: https://arxiv.org/abs/1708.08157 · HTML: https://mlsanity.com/paper/1708.08157/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2017.

## Built on by (most cited first, 30 of 41)

- [Self-Supervised Learning with Kernel Dependence Maximization](https://mlsanity.com/paper/2106.08320/) (2021)
- [RKHS-SHAP: Shapley Values for Kernel Methods](https://mlsanity.com/paper/2110.09167/) (2021)
- [Calibration tests beyond classification](https://mlsanity.com/paper/2210.13355/) (2022)
- [Calibration by Distribution Matching: Trainable Kernel Calibration Metrics](https://mlsanity.com/paper/2310.20211/) (2023)
- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [An Asymptotic Test for Conditional Independence using Analytic Kernel Embeddings](https://mlsanity.com/paper/2110.14868/) (2021)
- [Learning Counterfactually Invariant Predictors](https://mlsanity.com/paper/2207.09768/) (2022)
- [Dynamic Flows on Curved Space Generated by Labeled Data](https://mlsanity.com/paper/2302.00061/) (2023)
- [Doubly Robust Conditional Independence Testing with Generative Neural Networks](https://mlsanity.com/paper/2407.17694/) (2024)
- [On the Hardness of Conditional Independence Testing In Practice](https://mlsanity.com/paper/2512.14000/) (2025)
- [Maximum Mean Discrepancy on Exponential Windows for Online Change Detection](https://mlsanity.com/paper/2205.12706/) (2022)
- [Random Smoothing Regularization in Kernel Gradient Descent Learning](https://mlsanity.com/paper/2305.03531/) (2023)
- [Kernelized Cumulants: Beyond Kernel Mean Embeddings](https://mlsanity.com/paper/2301.12466/) (2023)
- [Nystr"om M-Hilbert-Schmidt Independence Criterion](https://mlsanity.com/paper/2302.09930/) (2023)
- [Amortized Linear-time Exact Shapley Value for Product-Kernel Methods](https://mlsanity.com/paper/2505.16516/) (2025)
- [Exact Distribution-Free Hypothesis Tests for the Regression Function of Binary Classification via Conditional Kernel Mean Embeddings](https://mlsanity.com/paper/2103.05126/) (2021)
- [Human-in-the-loop Handling of Knowledge Drift](https://mlsanity.com/paper/2103.14874/) (2021)
- [Learning Invariant Representations with Missing Data](https://mlsanity.com/paper/2112.00881/) (2021)
- [Understanding Deep Generative Models with Generalized Empirical Likelihoods](https://mlsanity.com/paper/2306.09780/) (2023)
- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Keep it Tighter -- A Story on Analytical Mean Embeddings](https://mlsanity.com/paper/2110.09516/) (2021)
- [Robust Bayesian Inference for Measurement Error Misspecification: The Berkson and Classical Cases](https://mlsanity.com/paper/2306.01468/) (2023)
- [The Minimax Rate of HSIC Estimation for Translation-Invariant Kernels](https://mlsanity.com/paper/2403.07735/) (2024)
- [Learning Representations for Independence Testing](https://mlsanity.com/paper/2409.06890/) (2024)
- [All Models Are Miscalibrated, But Some Less So: Comparing Calibration with Conditional Mean Operators](https://mlsanity.com/paper/2502.11465/) (2025)
- [Sequence Modeling with Spectral Mean Flows](https://mlsanity.com/paper/2510.15366/) (2025)
- [The Generalised Kernel Covariance Measure](https://mlsanity.com/paper/2604.03721/) (2026)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Bandits for Learning to Explain from Explanations](https://mlsanity.com/paper/2102.03815/) (2021)
- [Estimating R'enyi's α-Cross-Entropies in a Matrix-Based Way](https://mlsanity.com/paper/2109.11737/) (2021)

## Abstract (arXiv)

Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of characteristic property of the tensor product kernel.