Ah you're totally right about it not producing an exact copy, they even mention how they used different error functions for the different classes of distortion,

Ah you're totally right about it not producing an exact copy, they even mention how they used different error functions for the different classes of distortion, my mistake. But w.r.t. the regularization term, what I thought they meant by "we replace the regularizer R(x) with the implicit prior captured by the neural network" at the end of pg 2 was that they let the natural behavior of a neural network during optimization serve as the regularization, without need for an explicit regularization term. Not entirely sure though.