  Classical scalar-response regression methods treat covariates as a vector and
estimate a corresponding vector of regression coefficients. In medical
applications, however, regressors are often in a form of multi-dimensional
arrays. For example, one may be interested in using MRI imaging to identify
which brain regions are associated with a health outcome. Vectorizing the
two-dimensional image arrays is an unsatisfactory approach since it destroys
the inherent spatial structure of the images and can be computationally
challenging. We present an alternative approach - regularized matrix regression
- where the matrix of regression coefficients is defined as a solution to the
specific optimization problem. The method, called SParsity Inducing Nuclear
Norm EstimatoR (SpINNEr), simultaneously imposes two penalty types on the
regression coefficient matrix---the nuclear norm and the lasso norm---to
encourage a low rank matrix solution that also has entry-wise sparsity. A
specific implementation of the alternating direction method of multipliers
(ADMM) is used to build a fast and efficient numerical solver. 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.
