Kernelized Cumulants: Beyond Kernel Mean Embeddings
This work introduces a novel statistical framework for machine learning that generalizes existing kernel methods, offering potential improvements in data analysis tasks.
The paper tackles the problem of extending cumulants to reproducing kernel Hilbert spaces, showing they are computationally tractable via a kernel trick and provide new all-purpose statistics, with experiments on synthetic, environmental, and traffic data demonstrating advantages beyond degree one with similar computational complexity.
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.