Composition as Direction: An Active-Set Ray-Based Model for Sparse High-Dimensional Compositional Data

2026-06-27

AI Summary

This work addresses the challenge of modeling high-dimensional sparse compositional data, where exact zeros, component dependencies, and the unit-sum constraint coexist. The authors propose a novel framework that treats compositional observations as directions of latent abundance vectors lying in the non-negative orthant of the unit hypersphere. An active set process identifies non-zero components, and a latent Gaussian density is introduced along the positive rays of these active components to model the positive subcompositions. A radial variable further decouples direction from scale. This approach uniquely decomposes compositional modeling into two stages—active set selection and density estimation for positive subcompositions—while preserving a latent Gaussian structure that accommodates arbitrary dependencies. The method achieves both computational tractability and statistical flexibility, making it well-suited for high-dimensional sparse settings such as microbiome data.

Abstract

[Working Draft] Compositional data are central to microbial, ecological, and environmental research, yet often have four features that are difficult to accommodate jointly: exact zeros, latent dependence among components, high-dimensionality, and a unit-sum constraint that induces a non-Euclidean geometry. Conventional Dirichlet-type and logistic-normal models address these features only partially. Projected Gaussian models offer a directional representation that captures exact zeros and latent dependence; however, support correctness on the simplex requires either truncation or folding, both of which become computationally prohibitive as the dimension grows. We develop an Active-set Ray-based Compositional (ARC) framework, which retains the benefits of projected Gaussian models while remaining computationally feasible in high-dimensional settings. In this framework, we map compositions to the nonnegative orthant of the unit hypersphere and specify an active-set process that governs which components are present. Conditional on the active set, the positive subcomposition is modeled by evaluating a latent Gaussian density along positive rays of the active subspace with the radius treated as an auxiliary variable. Such a construction (i) separates the active-set process that governs which components are present from the positive subcomposition on the active components, (ii) preserves a latent Gaussian interpretation, and (iii) accommodates arbitrary latent dependence. Thus, the framework is conducive to high-dimensional applications in which exact zeros and shared positive responses are scientifically central. Conceptually, the proposed framework reframes a composition as an observed direction of a latent abundance vector with an unobserved magnitude and an explicitly modeled active set.

Problem

• compositional data
• exact zeros
• high-dimensionality
• latent dependence
• unit-sum constraint

Innovation

• compositional data
• active-set
• projected Gaussian
• high-dimensional
• exact zeros