  Connectivity estimation is challenging in the context of high-dimensional
data. A useful preprocessing step is to group variables into clusters, however,
it is not always clear how to do so from the perspective of connectivity
estimation. Another practical challenge is that we may have data from multiple
related classes (e.g., multiple subjects or conditions) and wish to incorporate
constraints on the similarities across classes. We propose a probabilistic
model which simultaneously performs both a grouping of variables (i.e.,
detecting community structure) and estimation of connectivities between the
groups which correspond to latent variables. The model is essentially a factor
analysis model where the factors are allowed to have arbitrary correlations,
while the factor loading matrix is constrained to express a community
structure. The model can be applied on multiple classes so that the
connectivities can be different between the classes, while the community
structure is the same for all classes. We propose an efficient estimation
algorithm based on score matching, and prove the identifiability of the model.
Finally, we present an extension to directed (causal) connectivities over
latent variables. Simulations and experiments on fMRI data validate the
practical utility of the method.
