  The study of hierarchy in networks of the human brain has been of significant
interest among the researchers as numerous studies have pointed out towards a
functional hierarchical organization of the human brain. This paper provides a
novel method for the extraction of hierarchical connectivity components in the
human brain using resting-state fMRI. The method builds upon prior work of
Sparse Connectivity Patterns (SCPs) by introducing a hierarchy of sparse
overlapping patterns. The components are estimated by deep factorization of
correlation matrices generated from fMRI. The goal of the paper is to extract
interpretable hierarchical patterns using correlation matrices where a low rank
decomposition is formed by a linear combination of a high rank decomposition.
We formulate the decomposition as a non-convex optimization problem and solve
it using gradient descent algorithms with adaptive step size. We also provide a
method for the warm start of the gradient descent using singular value
decomposition. We demonstrate the effectiveness of the developed method on two
different real-world datasets by showing that multi-scale hierarchical SCPs are
reproducible between sub-samples and are more reproducible as compared to
single scale patterns. We also compare our method with existing hierarchical
community detection approaches. Our method also provides novel insight into the
functional organization of the human brain.
