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We show that the influence of a subset of the training samples can be removed -- or "forgotten" -- from the weights of a network trained on large-scale image classification tasks, and we provide strong computable bounds on the amount of remaining information after forgetting. Inspired by real-world applications of forgetting techniques, we introduce a novel notion of forgetting in mixed-privacy setting, where we know that a "core" subset of the training samples does not need to be forgotten. While this variation of the problem is conceptually simple, we show that working in this setting significantly improves the accuracy and guarantees of forgetting methods applied to vision classification tasks. Moreover, our method allows efficient removal of all information contained in non-core data by simply setting to zero a subset of the weights with minimal loss in performance. We achieve these results by replacing a standard deep network with a suitable linear approximation. With opportune changes to the network architecture and training procedure, we show that such linear approximation achieves comparable performance to the original network and that the forgetting problem becomes quadratic and can be solved efficiently even for large models. Unlike previous forgetting methods on deep networks, ours can achieve close to the state-of-the-art accuracy on large scale vision tasks. In particular, we show that our method allows forgetting without having to trade off the model accuracy.