  Kernel methods have been among the most popular techniques in machine
learning, where learning tasks are solved using the property of reproducing
kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis
framework with reproducing kernel Hilbert $C^*$-module (RKHM) and kernel mean
embedding (KME) in RKHM. Since RKHM contains richer information than RKHS or
vector-valued RKHS (vv RKHS), analysis with RKHM enables us to capture and
extract structural properties in multivariate data, functional data and other
structured data. We show a branch of theories for RKHM to apply to data
analysis, including the representer theorem, and the injectivity and
universality of the proposed KME. We also show RKHM generalizes RKHS and vv
RKHS. Then, we provide concrete procedures for employing RKHM and the proposed
KME to data analysis.
