Robust Estimation of Tree Structured Gaussian Graphical Models
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 is independent, this allows us to 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 model. We prove the unidentifiability of this problem and provide the additional constraints to make this problem identifiable. We also provide an $\mathcal{O}(n^3)$ algorithm to find the underlying tree structure.