  Compositional data, such as human gut microbiomes, consist of non-negative
variables whose only the relative values to other variables are available.
Analyzing compositional data such as human gut microbiomes needs a careful
treatment of the geometry of the data. A common geometrical understanding of
compositional data is via a regular simplex. Majority of existing approaches
rely on a log-ratio or power transformations to overcome the innate simplicial
geometry. In this work, based on the key observation that a compositional data
are projective in nature, and on the intrinsic connection between projective
and spherical geometry, we re-interpret the compositional domain as the
quotient topology of a sphere modded out by a group action. This
re-interpretation allows us to understand the function space on compositional
domains in terms of that on spheres and to use spherical harmonics theory along
with reflection group actions for constructing a compositional Reproducing
Kernel Hilbert Space (RKHS). This construction of RKHS for compositional data
will widely open research avenues for future methodology developments. In
particular, well-developed kernel embedding methods can be now introduced to
compositional data analysis. The polynomial nature of compositional RKHS has
both theoretical and computational benefits. The wide applicability of the
proposed theoretical framework is exemplified with nonparametric density
estimation and kernel exponential family for compositional data.
