[nilkn]

I don't think I follow here. The goal of training the network isn't to encode the training data in the model, but rather to build a model that generalizes well. If the neural network has just memorized the training examples, then it overfit and really isn't useful in the real world.

I'm imagining a hypothetical example where generalization is easier to achieve with a neural network than with a decision tree using standard training techniques. Then a tree trained on the network might generalize better than a tree trained straight on the original data, with the additional benefit of being less of a black box than the network.