Title: NTK-SAP: Improving neural network pruning by aligning training dynamics. Abstract: Pruning neural networks before training has received increasing interest due to its potential to reduce training time and memory. One popular method is to prune the connections based on a  certain metric, but it is not entirely clear what metric is the best choice. Recent advances in neural tangent kernel (NTK) theory suggest that the training dynamics of large enough neural networks is closely related to the spectrum of the NTK. Motivated by this finding, we propose to prune the connections that have the least influence on the spectrum of the NTK. This method can help maintain the NTK spectrum, which may help align the training dynamics to that of its dense counterpart. However, one possible issue is that the fixed-weight-NTK corresponding to a given initial point can be very different from the NTK corresponding to later iterates during the training phase. We further propose to sample multiple realizations of random weights to estimate the NTK spectrum. Note that our approach is weight-agnostic, which is different from most existing methods that are weight-dependent. In addition, we use random inputs to compute the fixed-weight-NTK, making our method data-agnostic as well. We name our foresight pruning algorithm Neural Tangent Kernel Spectrum-Aware Pruning (NTK-SAP). Empirically, our method achieves better performance than all baselines on multiple datasets.