  A number of recent approaches have been proposed for pruning neural network
parameters at initialization with the goal of reducing the size and
computational burden of models while minimally affecting their training
dynamics and generalization performance. While each of these approaches have
some amount of well-founded motivation, a rigorous analysis of the effect of
these pruning methods on network training dynamics and their formal
relationship to each other has thus far received little attention. Leveraging
recent theoretical approximations provided by the Neural Tangent Kernel, we
unify a number of popular approaches for pruning at initialization under a
single path-centric framework. We introduce the Path Kernel as the
data-independent factor in a decomposition of the Neural Tangent Kernel and
show the global structure of the Path Kernel can be computed efficiently. This
Path Kernel decomposition separates the architectural effects from the
data-dependent effects within the Neural Tangent Kernel, providing a means to
predict the convergence dynamics of a network from its architecture alone. We
analyze the use of this structure in approximating training and generalization
performance of networks in the absence of data across a number of
initialization pruning approaches. Observing the relationship between input
data and paths and the relationship between the Path Kernel and its natural
norm, we additionally propose two augmentations of the SynFlow algorithm for
pruning at initialization.
