Testing properties of trees in graphical models with covariance queries

TL;DR

This paper develops efficient randomized tests for global structural properties of trees in high-dimensional graphical models using covariance queries, with explicit bounds on query complexity.

Contribution

It introduces novel testing procedures for fundamental tree properties that are more query-efficient than full reconstruction methods.

Findings

Tests for properties like number of leaves, maximum degree, and diameter are query-efficient.

Explicit bounds on query complexity depend on property thresholds and tolerances.

Certain global properties can be tested without reconstructing the entire tree.

Abstract

We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.