  Kernel mean embedding (KME) is a powerful tool to analyze probability
measures for data, where the measures are conventionally embedded into a
reproducing kernel Hilbert space (RKHS). In this paper, we generalize KME to
that of von Neumann-algebra-valued measures into reproducing kernel Hilbert
modules (RKHMs), which provides an inner product and distance between von
Neumann-algebra-valued measures. Von Neumann-algebra-valued measures can, for
example, encode relations between arbitrary pairs of variables in a
multivariate distribution or positive operator-valued measures for quantum
mechanics. Thus, this allows us to perform probabilistic analyses explicitly
reflected with higher-order interactions among variables, and provides a way of
applying machine learning frameworks to problems in quantum mechanics. We also
show that the injectivity of the existing KME and the universality of RKHS are
generalized to RKHM, which confirms many useful features of the existing KME
remain in our generalized KME. And, we investigate the empirical performance of
our methods using synthetic and real-world data.

Machine Learning;Machine Learning;Operator Algebras