Paper:
High resolution neural connectivity from incomplete tracing data using nonnegative spline regression
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 µm 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.

Review:
This paper presents a structured regression model for inferring neural connectivity from viral tracing data. Combining a nonnegativity constraint, a spatial smoothness regularizer, a low-rank assumption, and a mask that handles ambiguity within the injection site, the authors derive a model that can leverage observed tracer data to infer the underlying connectivity. While the individual components of this model have been explored in other structured regression problems, this paper presents a thorough and well-executed application of these ideas to an interesting problem.
The underlying model is a non-negative linear regression, y = Wx + \eta, where \eta is drawn from a spherical Gaussian model.  The weight matrix, W, is assumed to be nonnegative and, in probabilistic terms, drawn from a spatially smooth prior. Optionally, a low-rank assumption may be incorporated into the weight model, which can dramatically improve memory efficiency for large-scale problems.  While the individual components of this model (nonnegative regression, Laplacian regularized least squares, low-rank constraints) are well-studied, I think this is a nice combination and application of these techniques to a real-world, scientific problem. The presentation of the model, the synthetic examples, and the real world applications (and supplementary movies) are particularly clear.  While it is certainly valid to directly construct a objective function that captures both the reconstruction error and the domain-specific constraints and inductive biases, I think a probabilistic perspective could elucidate a number of potential extensions and connections to existing work. For example, Jonas and Kording [1] tackle a similar connectomics problem by formulating a probabilistic generative model in terms of latent variables of each neuron. While the setup is somewhat different (theirs is an unsupervised modeling problem whereas yours is framed as a regression problem), one could imagine a similar set of latent variables serving as a prior for your weight matrix.  Similarly, from a probabilistic perspective, the Gaussian noise model is not particularly realistic given that the fluorescence observations are nonnegative. Likewise, the spherical covariance seems likely to be a strong assumption. While these may justifiable and computationally advantageous simplifications, it is worth discussing. Finally, the orthogonal projector, P_{\Omega}, seems like a rather fancy way of saying that your loss function is only summing over observable voxels.  Minor comments: - The labels in Figure 3 are not very legible.  [1] Jonas, Eric, and Konrad Kording. "Automatic discovery of cell types and microcircuitry from neural connectomics." eLife 4 (2015): e04250.