  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.
