  Whole-brain neural connectivity data are now available from viral tracing
experiments, which reveal the connections between a source injection site and
elsewhere in the brain. These hold the promise of revealing spatial patterns of
connectivity throughout the mammalian brain. To achieve this goal, we seek to
fit a weighted, nonnegative adjacency matrix among 100 $\mu$m brain "voxels"
using viral tracer data. Despite a multi-year experimental effort, injections
provide incomplete coverage, and the number of voxels in our data is orders of
magnitude larger than the number of injections, making the problem severely
underdetermined. Furthermore, projection data are missing within the injection
site because local connections there are not separable from the injection
signal.
  We use a novel machine-learning algorithm to meet these challenges and
develop a spatially explicit, voxel-scale connectivity map of the mouse visual
system. Our method combines three features: a matrix completion loss for
missing data, a smoothing spline penalty to regularize the problem, and
(optionally) a low rank factorization. We demonstrate the consistency of our
estimator using synthetic data and then apply it to newly available Allen Mouse
Brain Connectivity Atlas data for the visual system. Our algorithm is
significantly more predictive than current state of the art approaches which
assume regions to be homogeneous. We demonstrate the efficacy of a low rank
version on visual cortex data and discuss the possibility of extending this to
a whole-brain connectivity matrix at the voxel scale.
