Robust estimation of tree structured Gaussian Graphical Model

TL;DR

This paper investigates the problem of recovering the structure of tree-structured Gaussian graphical models from noisy observations, demonstrating unidentifiability in general but providing conditions and algorithms for limited cases.

Contribution

It proves unidentifiability of the model structure under noise, characterizes when the structure can be identified, and offers an efficient algorithm for finding the equivalence class of trees.

Findings

Unidentifiability is limited to a small class of candidate trees.

Additional constraints can ensure identifiability.

An O(n^3) algorithm is provided for structure recovery.

Abstract

Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise at each node is independent, we can compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical models. We prove that this problem is unidentifiable, but show that this unidentifiability is limited to a small class of candidate trees. We further present additional constraints under which the problem is identifiable. Finally, we provide an O(n^3) algorithm to find this equivalence class of trees.