Title: Max-Affine Spline Insights Into Deep Network Pruning. Abstract: State-of-the-art (SOTA) approaches to deep network (DN) training overparametrize the model and then prune a posteriori to obtain a "winning ticket'' subnetwork that can be trained from scratch to achieve high accuracy. To date, the literature has remained largely empirical and hence provides little insights into how pruning affects a DN's decision boundary and no guidance regarding how to design a principled pruning technique. Using a recently developed spline interpretation of DNs, we develop new theory and visualization tools that provide new insights into how pruning DN nodes affects the decision boundary. We discover that a DN's spline mappings exhibit an early-bird (EB) phenomenon whereby the spline's partition converges at early training stages, bridging the recently developed max-affine spline theory and lottery ticket hypothesis of DNs. We leverage this new insight to develop a principled and efficient pruning strategy that focuses on a tiny fraction of DN nodes whose corresponding spline partition regions actually contribute to the final decision boundary. Extensive experiments on four networks and three datasets validate that our new spline-based DN pruning approach reduces training FLOPs by up to 3.5x while achieving similar or even better accuracy than state-of-the-art methods. All the codes will be released publicly upon acceptance.