  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....
