# Kernelized Cumulants: Beyond Kernel Mean Embeddings

arXiv: https://arxiv.org/abs/2301.12466 · HTML: https://mlsanity.com/paper/2301.12466/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 57 references that are in the map.
- First posted on arXiv 29 Jan 2023.

## Builds on (most cited first, 8 of 8)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A Survey on Graph Kernels](https://mlsanity.com/paper/1903.11835/) (2019)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Characteristic and Universal Tensor Product Kernels](https://mlsanity.com/paper/1708.08157/) (2017)
- [Kernel-based Tests for Joint Independence](https://mlsanity.com/paper/1603.00285/) (2016)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [A Kernel Test for Three-Variable Interactions](https://mlsanity.com/paper/1306.2281/) (2013)

## Built on by (most cited first, 5 of 5)

- [A Unified Data Representation Learning for Non-parametric Two-sample Testing](https://mlsanity.com/paper/2412.00613/) (2024)
- [Information-Theoretic Measures on Lattices for High-Order Interactions](https://mlsanity.com/paper/2408.07533/) (2024)
- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)
- [Kernel Integrated R2: A Measure of Dependence](https://mlsanity.com/paper/2602.22985/) (2026)

## Abstract (arXiv)

In mathbb Rd, it is well-known that cumulants provide an alternative to moments that can achieve the same goals with numerous benefits such as lower variance estimators. In this paper we extend cumulants to reproducing kernel Hilbert spaces (RKHS) using tools from tensor algebras and show that they are computationally tractable by a kernel trick. These kernelized cumulants provide a new set of all-purpose statistics; the classical maximum mean discrepancy and Hilbert-Schmidt independence criterion arise as the degree one objects in our general construction. We argue both theoretically and empirically (on synthetic, environmental, and traffic data analysis) that going beyond degree one has several advantages and can be achieved with the same computational complexity and minimal overhead in our experiments.