  The problem of linking functional connectomics to behavior is extremely
challenging due to the complex interactions between the two distinct, but
related, data domains. We propose a coupled manifold optimization framework
which projects fMRI data onto a low dimensional matrix manifold common to the
cohort. The patient specific loadings simultaneously map onto a behavioral
measure of interest via a second, non-linear, manifold. By leveraging the
kernel trick, we can optimize over a potentially infinite dimensional space
without explicitly computing the embeddings. As opposed to conventional
manifold learning, which assumes a fixed input representation, our framework
directly optimizes for embedding directions that predict behavior. Our
optimization algorithm combines proximal gradient descent with the trust region
method, which has good convergence guarantees. We validate our framework on
resting state fMRI from fifty-eight patients with Autism Spectrum Disorder
using three distinct measures of clinical severity. Our method outperforms
traditional representation learning techniques in a cross validated setting,
thus demonstrating the predictive power of our coupled objective.
