Title: 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 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. more »« less

Tugnait, Jitendra K.(
, 2020 IEEE 30th International Workshop on Machine Learning for Signal Processing (MLSP))

null
(Ed.)

We consider the problem of estimating the conditional independence graph (CIG) of a sparse, high-dimensional proper complex-valued Gaussian graphical model (CGGM). For CGGMs, the problem reduces to estimation of the inverse covariance matrix with more unknowns than the sample size. We consider a smoothly clipped absolute deviation (SCAD) penalty instead of the ℓ 1 -penalty to regularize the problem, and analyze a SCAD-penalized log-likelihood based objective function to establish consistency and sparsistency of a local estimator of inverse covariance in a neighborhood of the true value. A numerical example is presented to illustrate the advantage of SCAD-penalty over the usual ℓ 1 -penalty.

We consider the problem of estimating differences in two Gaussian graphical models (GGMs) which are known to have similar structure. The GGM structure is encoded in its precision (inverse covariance) matrix. In many applications one is interested in estimating the difference in two precision matrices to characterize underlying changes in conditional dependencies of two sets of data. Most existing methods for differential graph estimation are based on a lasso penalized loss function. In this paper, we analyze a log-sum penalized D-trace loss function approach for differential graph learning. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function. Theoretical analysis establishing consistency in estimation in high-dimensional settings is provided. We illustrate our approach using a numerical example where log-sum penalized D-trace loss significantly outperforms lasso-penalized D-trace loss as well as smoothly clipped absolute deviation (SCAD) penalized D-trace loss.

Motivated by modern applications in which one constructs graphical models based on a very large number of features, this paper introduces a new class of cluster-based graphical models, in which variable clustering is applied as an initial step for reducing the dimension of the feature space. We employ model assisted clustering, in which the clusters contain features that are similar to the same unobserved latent variable. Two different cluster-based Gaussian graphical models are considered: the latent variable graph, corresponding to the graphical model associated with the unobserved latent variables, and the cluster-average graph, corresponding to the vector of features averaged over clusters. Our study reveals that likelihood based inference for the latent graph, not analyzed previously, is analytically intractable. Our main contribution is the development and analysis of alternative estimation and inference strategies, for the precision matrix of an unobservable latent vector Z. We replace the likelihood of the data by an appropriate class of empirical risk functions, that can be specialized to the latent graphical model and to the simpler, but under-analyzed, cluster-average graphical model. The estimators thus derived can be used for inference on the graph structure, for instance on edge strength or pattern recovery. Inference is based on the asymptotic limits of the entry-wise estimates of the precision matrices associated with the conditional independence graphs under consideration. While taking the uncertainty induced by the clustering step into account, we establish Berry-Esseen central limit theorems for the proposed estimators. It is noteworthy that, although the clusters are estimated adaptively from the data, the central limit theorems regarding the entries of the estimated graphs are proved under the same conditions one would use if the clusters were known in advance. As an illustration of the usage of these newly developed inferential tools, we show that they can be reliably used for recovery of the sparsity pattern of the graphs we study, under FDR control, which is verified via simulation studies and an fMRI data analysis. These experimental results confirm the theoretically established difference between the two graph structures. Furthermore, the data analysis suggests that the latent variable graph, corresponding to the unobserved cluster centers, can help provide more insight into the understanding of the brain connectivity networks relative to the simpler, average-based, graph.

Summary For multivariate spatial Gaussian process models, customary specifications of cross-covariance functions do not exploit relational inter-variable graphs to ensure process-level conditional independence between the variables. This is undesirable, especially in highly multivariate settings, where popular cross-covariance functions, such as multivariate Matérn functions, suffer from a curse of dimensionality as the numbers of parameters and floating-point operations scale up in quadratic and cubic order, respectively, with the number of variables. We propose a class of multivariate graphical Gaussian processes using a general construction called stitching that crafts cross-covariance functions from graphs and ensures process-level conditional independence between variables. For the Matérn family of functions, stitching yields a multivariate Gaussian process whose univariate components are Matérn Gaussian processes, and which conforms to process-level conditional independence as specified by the graphical model. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate the utility of the graphical Matérn Gaussian process to jointly model highly multivariate spatial data using simulation examples and an application to air-pollution modelling.

Tugnait, Jitendra K.(
, 2022 56th Asilomar Conference on Signals, Systems, and Computers)

We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary, multivariate long-range dependent (LRD) Gaussian time series. In a time series graph, each component of the vector series is represented by a distinct node, and associations between components are represented by edges between the corresponding nodes. In a recent work on graphical modeling of short-range dependent (SRD) Gaussian time series, the problem was cast as one of multi-attribute graph estimation for random vectors where a vector is associated with each node of the graph. At each node, the associated random vector consists of a time series component and its delayed copies. A theoretical analysis based on short-range dependence has been given in Tugnait (2022 ICASSP). In this paper we analyze this approach for LRD Gaussian time series and provide consistency results regarding convergence in the Frobenius norm of the inverse covariance matrix associated with the multi-attribute graph.

Katiyar, Ashish, Hoffmann, Jessica, and Caramanis, Constantine. Robust Estimation of Tree Structured Gaussian Graphical Models. Retrieved from https://par.nsf.gov/biblio/10173879. Proceedings of Machine Learning Research 97.

Katiyar, Ashish, Hoffmann, Jessica, and Caramanis, Constantine.
"Robust Estimation of Tree Structured Gaussian Graphical Models". Proceedings of Machine Learning Research 97 (). Country unknown/Code not available. https://par.nsf.gov/biblio/10173879.

@article{osti_10173879,
place = {Country unknown/Code not available},
title = {Robust Estimation of Tree Structured Gaussian Graphical Models},
url = {https://par.nsf.gov/biblio/10173879},
abstractNote = {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.},
journal = {Proceedings of Machine Learning Research},
volume = {97},
author = {Katiyar, Ashish and Hoffmann, Jessica and Caramanis, Constantine},
}

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