Structure Learning for Directed Trees with Zero-Inflated Compositional Nodes

TL;DR

This paper presents a new method for learning directed tree structures among compositional data vectors, effectively handling zero-inflation and respecting simplex geometry, with proven consistency and practical applications in microbiome and single-cell data.

Contribution

It introduces a novel framework for directed tree structure learning over compositional nodes using KL divergence, ensuring identifiability and consistency.

Findings

Method accurately recovers true structures in simulations.

Applied to microbiome and single-cell data, revealing biologically meaningful directions.

Finite-sample guarantees relate sample size to signal strength and data dimensions.

Abstract

Compositional data, which are vectors of proportions constrained to the probability simplex, arise frequently in modern scientific applications, including microbiome relative abundances across body sites and cell-type mixture weights derived from single-cell genomics. While regression methods for compositional data are well developed, no existing graphical model framework addresses the problem of learning conditional dependence structures among multiple compositional vectors. This paper introduces a novel framework for directed tree structure learning over compositional nodes. We employ the Kullback-Leibler divergence as the scoring function and model the conditional expectation of each child composition as a mixture of a baseline composition and a parent-driven component parameterized by a column-stochastic transition matrix. This formulation respects the simplex geometry, handles zero-inflated compositions gracefully, and, combined with a non-degeneracy condition on the transition matrix, ensures identifiability of edge directions from observational data. We prove consistency of structure recovery and derive finite-sample guarantees that characterize the required sample size in terms of the signal gap, node dimension, and penalty level. The efficacy of our approach is demonstrated through simulations and applications to multi-site microbiome data and single-cell data, yielding interpretable directed structures that align with known biological mechanisms.