Algebraic tests of general Gaussian latent tree models

TL;DR

This paper characterizes the covariance matrices of general Gaussian latent tree models using polynomial constraints, and introduces a new testing method that overcomes computational challenges and singularity issues, applicable to models with many variables.

Contribution

It provides a full semi-algebraic description of Gaussian latent tree models with observed variables not limited to leaves, and proposes a novel testing approach that handles complex constraints and singularities.

Findings

New polynomial constraints characterize covariance matrices in latent tree models.

The proposed test avoids likelihood maximization and manages high-dimensional constraints.

Numerical experiments demonstrate the effectiveness of the testing methodology.

Abstract

We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any such model. In other words, we find polynomial constraints that characterize when a matrix is the covariance matrix of a distribution in a given latent tree model. However, leveraging these constraints to test a given such model is often complicated by the number of constraints being large and by singularities of individual polynomials, which may invalidate standard approximations to relevant probability distributions. Illustrating with the star tree, we propose a new testing methodology that circumvents singularity issues by trading off some statistical estimation efficiency and handles cases with many constraints through recent advances on Gaussian approximation for maxima of sums of high-dimensional random vectors. Our test avoids the need to maximize the possibly multimodal likelihood function of such models and is applicable to models with larger number of variables. These points are illustrated in numerical experiments.