Reproducing Kernels and New Approaches in Compositional Data Analysis

Binglin Li, Jeongyoun Ahn

Abstract:

Compositional data, such as human gut microbiomes, consist of non-negative variables
where only the relative values of these variables are available. Analyzing compositional
data requires careful treatment of the geometry of the data. A common geometrical approach to understanding such data is through a regular simplex. The majority of existing
approaches rely on log-ratio or power transformations to address the inherent simplicial
geometry. In this work, based on the key observation that compositional data are projective, we reinterpret the compositional domain as a group quotient of a sphere, leveraging
the intrinsic connection between projective and spherical geometry. This interpretation
enables us to understand the function spaces on the compositional domain in terms of
those on a sphere, and furthermore, to utilize spherical harmonics theory for constructing
a compositional Reproducing Kernel Hilbert Space (RKHS). The construction of RKHS for
compositional data opens up new research avenues for future methodology developments,
particularly introducing well-developed kernel methods to compositional data analysis. We
demonstrate the wide applicability of the proposed theoretical framework with examples of
nonparametric density estimation, kernel exponential family, and support vector machine
for compositional data.