[Paper Review] Reproducing Kernels and New Approaches in Compositional Data Analysis

Binglin Li, Jeongyoun Ahn|arXiv (Cornell University)|May 2, 2022

Geochemistry and Geologic Mapping4 citations

TL;DR

This paper introduces a novel Reproducing Kernel Hilbert Space (RKHS) framework for compositional data by modeling the simplex domain as a quotient of the sphere under a group action, enabling kernel-based methods without log-ratio transformations. The approach leverages spherical harmonics and reflection group symmetries to construct polynomial kernels, yielding a theoretically grounded, computationally efficient RKHS that supports nonparametric density estimation and kernel exponential families on compositional data.

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.

Motivation & Objective

• To address the limitations of traditional log-ratio and power transformations in compositional data analysis by developing a geometry-aware kernel framework.
• To establish a theoretical foundation for kernel methods on compositional data by reinterpreting the simplex as a quotient space of the sphere under group action.
• To construct a Reproducing Kernel Hilbert Space (RKHS) for compositional data using spherical harmonics and reflection group symmetries.
• To enable advanced statistical methods—such as nonparametric density estimation and kernel exponential families—on compositional data through kernel embedding techniques.
• To provide a polynomial, computationally tractable kernel structure that avoids the instability and sensitivity associated with zero-handling in log-ratio methods.

Proposed method

• Reinterpret the compositional domain Δ^d as the quotient space S^d / Γ, where Γ is a group of reflections, leveraging projective and spherical geometry.
• Construct a compositional RKHS using spherical harmonics of even degrees, ensuring invariance under the group action Γ.
• Define projective kernels via the inner product of normalized spherical vectors: k^p_m(y_i, y_j) = (1 + y_i·y_j)^{2m} for y_i, y_j ∈ S^d, y_i ≠ -y_j.
• Utilize the kernel mean embedding approach to represent probability distributions as elements in the RKHS, avoiding direct estimation of mean points on non-linear manifolds.
• Apply representer theorems to show that solutions to interpolation and regularization problems lie in finite-dimensional subspaces spanned by kernel functions at observed data points.
• Ensure linear independence of kernel functions for large m by proving full rank of the Gram matrix, enabling stable numerical computation.

Experimental results

Research questions

• RQ1 How can a Reproducing Kernel Hilbert Space (RKHS) be constructed for compositional data that respects the intrinsic geometry of the simplex without relying on log-ratio transformations?
• RQ2 What is the role of spherical harmonics and reflection group actions in building a symmetric, polynomial kernel on the projective sphere S^d / Γ?
• RQ3 Can kernel mean embedding be effectively applied to compositional data by embedding distributions into an RKHS defined on the quotient space?
• RQ4 How do the proposed kernels support nonparametric density estimation and kernel exponential families on compositional domains?
• RQ5 What theoretical guarantees (e.g., representer theorems) can be established for interpolation and regularization in the proposed compositional RKHS?

Key findings

• The proposed kernel construction ensures that the Gram matrix of the projective kernel is diagonally dominant and full rank for sufficiently large m, guaranteeing linear independence of the kernel functions.
• The minimal norm interpolation problem in the compositional RKHS admits a unique solution expressible as a linear combination of kernel functions centered at the data points, as proven by the representer theorem.
• The regularization problem for compositional data admits a unique solution in the form of a linear combination of kernel functions, with coefficients determined by a system of linear equations derived from the kernel and data constraints.
• The kernel mean embedding approach provides a distributional alternative to pointwise mean estimation, avoiding the geometric inconsistency of defining means on non-linear spaces like the simplex.
• The framework enables nonparametric density estimation and kernel exponential family modeling on compositional data by leveraging the rich structure of the constructed RKHS.
• The polynomial nature of the kernel ensures both theoretical tractability and computational efficiency, avoiding the numerical instability of zero-imputation in log-ratio methods.