Reproducing Kernels and New Approaches in Compositional Data Analysis

Abstract

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.

Supervised learning and model analysis with compositional data

“…Our proposed framework extends these works by providing new post-analysis techniques (e.g., the compositional feature influence) that respect the compositional structure. Recently, [25,38] used the radial transformation to argue that kernels on the sphere provide a natural way of analyzing compositions with zeros and similar to our work suggest using the kernel embeddings in a subsequent analysis. Part (ii) is related to the fields of explainable artificial intelligence [39] and interpretable machine learning [40], which focus on extracting information from predictive models.…”