One-step neural network inversion with PDF learning and emulation

We present two new types of neural networks (both of which can be trained with ordinary error backpropagation) and we present a new algorithm for learning a probability density function (pdf) from example vectors. It is normally difficult to invert a neural network, but for the new bijective neural network, it is efficient to find an input producing any desired output, and such an input is guaranteed to exist and to be unique. Furthermore, it can be used as one component in building a pdf neural network, which is a neural network with a nonnegative output, and for which it is guaranteed that the integral of the output is exactly 1.0 (as in a pdf function). Both of these can be used for supervised learning using standard error backpropagation. Finally, the new pdf learning algorithm is capable of using those networks to learn a pdf given i.i.d. samples drawn from that pdf, and to then generate new vectors from the learned pdf. This, in turn, allows inversion of a function with non-unique inverses, where each inverse is generated with just a single evaluation of the network.