Kernelized Cumulants: Beyond Kernel Mean Embeddings

2023/01/29 by Patric Bonnier, Bonnier, Patric, Harald Oberhauser +3 · 1 citation
Mathematics · #FOS: Computer and information sciences #Information Theory (cs.IT) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Statistical Methods and Inference

paper · doi:10.48550/arxiv.2301.12466

openalex publication_date 2023/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In \mathbb R^d, 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.

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