Title: Getting away with more network pruning: From sparsity to geometry and linear regions. Abstract: One surprising trait of neural networks is the extent to which their connections can be pruned with little to no effect on accuracy. But when we cross a critical level of parameter sparsity, pruning any further leads to a sudden drop in accuracy. What could explain such a drop? In this work, we explore how sparsity may affect the geometry of the linear regions defined by a neural network and consequently reduce its expected maximum number of linear regions. We observe that sparsity affects accuracy in pruned neural networks similarly to how it affects the number of linear regions as well as - and more so - our proposed upper bound on that number. Conversely, we find out that selecting the sparsity on each layer to maximize the bound very often improves accuracy in comparison to using the same sparsity across all layers, thereby providing us guidance on where to prune.  