Title: Weighted Regularization for Efficient Neural Network Compression. Abstract: Regularization is widely applied to model complexity reduction and neural network compression. Existing $L_1$ and nuclear norm regularizations can achieve favorable results, but these methods treat all parameters equally and ignore the importance of the parameters. Taking the trained parameters as prior information to construct weights, a weighted regularization method is proposed in this paper. Theoretically, we establish the bounds on the estimation errors for values of the global minimum for a fully connected single hidden layer neural network. Further we prove the estimates generated from the weighted $L_1$ regularization and the weighted nuclear norm regularization can recover the sparsity and the low rank structure of a global minimum of the neural network with a high probability, respectively. The effectiveness of the algorithm is validated by conducting a numerical simulation and experiments with popular neural networks on public datasets from real-world applications.