A Sparsity Inducing Nuclear-Norm Estimator (SpINNEr) for Matrix-Variate Regression in Brain Connectivity Analysis

For example, it is of clinical interest to understand associations between: (a) alcoholism and the electrical activity of different brain regions over time collected from electroencephalography (EEG) (Li et al., 2010); (b) cognitive function and three-dimensional white-matter structure data collected from diffusion tensor imaging (DTI) (Goldsmith et al., 2014) for patients with multiple sclerosis (MS); and (c) cognitive impairment and brain's metabolic activity data collected from three-dimensional positron emission tomography (PET) imaging (Wang et al., 2014). Our work focuses on the problem of identifying brain network connections that are associated with neurocognitive measures for HIVinfected individuals. The outcome (response) is a continuous variable and the predictors are matrix representations of functional connectivity between the brain's cortical regions. Biophysical considerations motivate our interest in estimating a matrix of regression coefficients that has the following two properties: (i) it should be relatively sparse, since we aim to identify connections that most strongly predict the outcome; and more importantly, (ii) the response-related connections form clusters, since brain activity networks are known to consist of densely connected regions. These two properties translate to the coefficient matrix having relatively small clusters, or blocks of nonzero entries, which implies that it is low-rank. Hence, we aim to solve the matrix regression problem by estimating a coefficient matrix that is both sparse and low-rank. To further illustrate our approach, consider the three matrices in Figure 1. The one in the left panel is sparse, but full-rank, the one on the right panel is low-rank, but not sparse, while the one in the middle panel is both low-rank and sparse, which is the structure we are interested in. To find such a solution, we propose a regularization method called SParsity Inducing Nuclear Norm EstimatoR (SpINNEr).