Paper:
Network pruning is a promising avenue for compressing deep neural networks. A typical approach to pruning starts by training a model and removing redundant parameters while minimizing the impact on what is learned. Alternatively, a recent approach shows that pruning can be done at initialization prior to training, based on a pruning criterion called connection sensitivity. However, it remains unclear exactly why pruning an untrained, randomly initialized neural network is effective. In this work, by noting connection sensitivity as a form of gradients, we formally characterize initialization conditions to ensure reliable connection sensitivity measurements, which in turn yields effective pruning results. Moreover, we analyze the signal propagation properties of the resulting pruned networks and introduce a simple, data-free method to improve their trainability. Our modifications to the existing pruning at initialization method lead to improved results on all tested network models for image classification tasks. Furthermore, we empirically study the effect of supervision for pruning and demonstrate that our signal propagation perspective, combined with unsupervised pruning, can be useful in various scenarios where pruning is applied to non-standard arbitrarily-designed architectures.

Review:
This paper analyzes how signals propagate through randomly initialized neural networks that have undergone a kind of pruning/sparsification. The pruning method utilizes a metric called 'connection sensitivity', which has been used in prior work and which measures the infinitessimal impact of turning off specific parameters. The distribution of singular values in the layer-to-layer Jacobian matrices for pruned networks becomes increasingly pathological as the depth increases. This observation motivates the concept of 'layerwise dynamical isometry' (LDI), a slight generalization of the concept of 'dynamical isometry' that has been studied in prior work. Several methods for approximately obtaining differing amounts of LDI are investigated in a series of in-depth experiments that show a strong correlation between increased signal propagation and improved trainability of sparse networks.
Although there have been numerous works studying dynamical isometry as a principle for initializing very deep networks, and many other works studying pruning methods after (and recently before) training, as far as I'm aware there has been no prior work that examines the intersection of these two directions. As such, I found the contributions of this paper to be novel and believe the results will be of interest to practitioners and theorists alike.
An important contribution of this paper is in identifying that a main difficulty in pruning networks at initialization comes from degradation of signal propagation, leading to poor or impossible training. The numerous well-thought-out experiments provide compelling evidence that the trainability of pruned networks is highly correlated with spectral measures of the networks' Jacobians. 
The authors take this observation a step further by introducing a method for correcting the poor conditioning that can result from pruning. They show that enforcing Approximate Isometry on weights of the pruned connectivity pattern enables the pruned models to train much faster and often achieve better performance.
Finally, the authors look at two natural extensions of their analysis  to designing new high-performing architectures to situations where labels are not present. Overall, I found this paper to have a detailed and thorough experimental analysis and to present nice new perspectives on pruning and signal propagation.