Matrix-Variate Regression for Sparse, Low-Rank Estimation of Brain Connectivities Associated With a Clinical Outcome

Objective: We address the problem of finding brain connectivities that are associated with a clinical outcome or phenotype. Methods: The proposed framework regresses a (scalar) clinical outcome on matrix-variate predictors which arise in the form of brain connectivity matrices. For example, in a large cohort of subjects we estimate those regions of functional connectivities that are associated with neurocognitive scores. We approach this high-dimensional yet highly structured estimation problem by formulating a regularized estimation process that results in a low-rank coefficient matrix having a sparse set of nonzero entries which represent regions of biologically relevant connectivities. In contrast to the recent literature on estimating a sparse, low-rank matrix from a single noisy observation, our scalar-on-matrix regression framework produces a data-driven extraction of structures that are associated with a clinical response. The method, called Sparsity Inducing Nuclear-Norm Estimator (SpINNEr), simultaneously constrains the regression coefficient matrix in two ways: a nuclear norm penalty encourages low-rank structure while anℓ₁ norm encourages entry-wise sparsity. Results: Our simulations show that SpINNEr outperforms other methods in estimation accuracy when the response-related entries (representing the brain's functional connectivity) are arranged in well-connected communities. SpINNEr is applied to investigate associations between HIV-related outcomes and functional connectivity in the human brain. Conclusion and Significance: Overall, this work demonstrates the potential of SpINNEr to recover sparse and low-rank estimates under scalar-on-matrix regression framework.