Recent advances in neural network pruning have shown how it is possible to reduce the computational costs and memory demands of deep learning models before training. We focus on this framework and propose a new pruning at initialization algorithm that leverages the Neural Tangent Kernel (NTK) theory to align the training dynamics of the sparse network with that of the dense one. Specifically, we show how the usually neglected data-dependent component in the NTK's spectrum can be taken into account by providing an analytical upper bound to the NTK's trace obtained by decomposing neural networks into individual paths. This leads to our Path eXclusion (PX), a foresight pruning method designed to preserve the parameters that mostly influence the NTK's trace. PX is able to find lottery tickets (i.e. good paths) even at high sparsity levels and largely reduces the need for additional training. When applied to pre-trained models it extracts subnetworks directly usable for several downstream tasks, resulting in performance comparable to those of the dense counterpart but with substantial cost and computational savings.