  There is increasing interest in learning a set of small outcome-relevant
subgraphs in network-predictor regression. The extracted signal subgraphs can
greatly improve the interpretation of the association between the network
predictor and the response. In brain connectomics, the brain network for an
individual corresponds to a set of interconnections among brain regions and
there is a strong interest in linking the brain connectome to human cognitive
traits. Modern neuroimaging technology allows a very fine segmentation of the
brain, producing very large structural brain networks. Therefore, accurate and
efficient methods for identifying a set of small predictive subgraphs become
crucial, leading to discovery of key interconnected brain regions related to
the trait and important insights on the mechanism of variation in human
cognitive traits. We propose a symmetric bilinear model with $L_1$ penalty to
search for small clique subgraphs that contain useful information about the
response. A coordinate descent algorithm is developed to estimate the model
where we derive analytical solutions for a sequence of conditional convex
optimizations. Application of this method on human connectome and language
comprehension data shows interesting discovery of relevant interconnections
among several small sets of brain regions and better predictive performance
than competitors.
