Robust estimation of tree structured Gaussian Graphical Model
This addresses a robustness issue in graphical model estimation for researchers in statistics and machine learning, though it is incremental as it builds on known identifiability challenges.
The paper tackles the problem of recovering the conditional independence structure of a tree-structured Gaussian graphical model from noisy observations where noise is independent per node, proving that the problem is generally unidentifiable but limited to a small class of candidate trees, and provides an O(n^3) algorithm to find this equivalence class.
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.