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

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

1. Problem Statement

Compositional data, prevalent in microbial, ecological, and environmental research, present four simultaneous challenges: exact zeros, latent dependence among components, high dimensionality, and the unit-sum constraint (simplex closure). The unit-sum constraint induces a non-Euclidean geometry where observed proportions compete, often obscuring or reversing the sign of underlying latent associations driven by shared environmental factors.

Existing approaches fail to address these features jointly:

• Dirichlet-type models impose inflexible covariance structures tied to the mean and cannot represent arbitrary positive correlations induced by shared latent drivers.
• Logistic-normal models offer flexibility via log-ratio transformations but are undefined at exact zeros, requiring ad hoc replacements that conflate presence with dominance.
• Projected Gaussian models (e.g., Elliptically Symmetric Angular Gaussian) preserve directional structure and handle zeros but require conditioning on the nonnegative orthant. This necessitates evaluating a parameter-dependent orthant probability (the likelihood of a Gaussian vector landing in the positive orthant), which becomes computationally prohibitive as dimension $d$ increases due to the exponential divergence of variance in standard estimators.

2. Methodology: The Active-Set Ray-Based Compositional (ARC) Framework

The authors propose the ARC framework, which reframes a composition not as a point on the simplex, but as the observed direction of a latent abundance vector with an unobserved magnitude and an explicitly modeled active set.

Geometric Embedding:
The composition $u$ is mapped to the nonnegative orthant of the unit hypersphere $S^{d-1}_{0,+}$ via an elementwise square root transformation $y = \sqrt{u}$. This preserves exact zeros as boundary points ($u_k=0 \iff y_k=0$) without requiring replacement or regularization.

Active-Set Factorization:
The model decomposes the likelihood into two distinct processes:

1. The Active-Set Process ($A_i$): A discrete presence/absence process governing which components are present ($y_{ik} > 0$). This is modeled using a probit factor specification with covariates and latent factors to capture co-occurrence patterns. Structural zeros are treated as a modeled event rather than a nuisance.
2. The Positive Subcomposition ($y^+_i$): Conditional on the active set $A_i$, the positive subcomposition is modeled by evaluating a latent Gaussian density along the positive rays of the active subspace.

Latent Radius Augmentation:
A key innovation is the introduction of a latent radius $r_i > 0$. The latent active vector is defined as $x^+_i = r_i y^+_i$.

• The vector $x^+_i$ follows a multivariate Gaussian distribution $N(\mu^+_i, V_{A_i A_i})$ in the active subspace.
• The observed data $y^+_i$ is the direction of $x^+_i$.
• By treating $r_i$ as an auxiliary variable, the model evaluates the Gaussian density along the observed ray. This construction eliminates the need for the orthant normalizing constant (the probability that $x^+_i$ falls in the positive orthant) because the conditioning on the active set and the positivity of $r_i$ ensure the support is naturally restricted to the positive orthant by construction.

Covariance Structure:
To ensure scalability in high dimensions ($d \gg n$), the latent covariance $V$ is parameterized using a low-rank factor model: $V = \Lambda\Lambda' + D$. This reduces the parameter count from $O(d^2)$ to $O(dq + d)$ and allows for the recovery of arbitrary latent dependence (positive or negative) on the abundance scale, distinct from the compositional scale.

Inference:
The framework is implemented via a Bayesian hierarchical model using MCMC. The sampler utilizes:

• Latent radius updates (1D Metropolis).
• Albert-Chib augmentation for the probit active-set model.
• Conditionally Gaussian updates for factors, regression coefficients, and loadings.
  The computational cost per iteration scales linearly with $d$ for fixed factor rank and sparse active sets, avoiding the exponential cost of orthant probability evaluation.

3. Key Contributions

The paper claims four primary contributions:

1. Model Proposal: The ARC model for sparse, high-dimensional compositional data, deriving a likelihood that treats the radius as an auxiliary latent variable.
2. Structural Zero Handling: An extension to structural zeros via an active-set formulation that separates the presence/absence process from the positive subcomposition, allowing distinct covariates and priors for each.
3. Tractable Inference: The introduction of a low-rank factor covariance paired with a probit active-set model, yielding conditionally Gaussian updates that scale to high dimensions without orthant probability calculations.
4. Joint Recovery: Demonstration that the active-set process and latent positive dependence can be recovered jointly on sparse, high-dimensional data.

4. Results

A simulation study with $n=350$ and dimensions $d \in \{20, \dots, 1000\}$ under varying sparsity regimes and dependence structures (block and unstructured) yielded the following:

• Active-Set Recovery: The model consistently recovered the structure of co-occurrence (AUC 0.83–0.86) and pairwise co-activation rates (correlation > 0.97) across all dimensions and sparsity levels.
• Latent Dependence Recovery: The model successfully recovered the true latent correlation matrix $R$ (correlation between estimated and true correlations > 0.93, often > 0.99).
• Comparison to Baselines: The empirical correlation of observed compositions (the "observed baseline") failed to capture latent dependence, with correlations dropping to as low as 0.21 under high sparsity. This confirms that simplex closure severely obscures latent positive dependence, a signal the ARC model successfully recovers.
• Scalability: The method remained stable and accurate as $d$ increased, benefiting from the "blessing of dimensionality" where more components loading on the same factors sharpen the estimation of the shared subspace.

5. Significance and Claims

The paper positions the ARC framework as a conceptual and computational advancement for compositional data analysis.

• Conceptual Reframing: It redefines a composition as an observed direction of a latent abundance vector. This separation allows the model to distinguish between mechanisms governing presence (structural zeros, dispersal) and mechanisms governing dominance (competition, shared drivers).
• Overcoming Computational Barriers: By avoiding the orthant normalizing constant, the framework makes projected Gaussian models feasible for high-dimensional settings where previous truncation-based approaches were computationally intractable.
• Scientific Utility: The framework is particularly suited for fields like microbiome and ecology where exact zeros and shared positive responses are central scientific features rather than data artifacts. It recovers the "abundance-scale" dependence that is otherwise masked by the unit-sum constraint, providing a more accurate representation of biological and ecological processes.

The authors note that while the current implementation uses a fixed factor rank, the framework is compatible with adaptive rank selection (e.g., multiplicative gamma processes) and can be extended to spatial, spatio-temporal, and detection-error settings by leveraging the conditional Gaussian structure of the augmented active vectors.