Designing GANs: A Likelihood Ratio Approach

In the first set the samples follow the origin probability density hp Z q and in the second the target density f p X q . The target density f p X q is considered unknown while hpZ q can either be known with the possibility to produce samples Z j every time it is necessary or unknown in which case we have a second fixed training set t Z ju . Our goal is to design a deterministic transformation GpZ q so that the data t Y ju produced by applying the transformation Y " Gp Z q onto t Z ju follow the target density f p Y q . Of course one may wonder whether the proposed problem enjoys any solution, namely, whether there indeed exists a transformation GpZ q capable of transforming Z into Y with the former following the origin density hp Z q and the latter the target density f pY q . The problem of transforming random vectors has been analyzed by (Box & Cox, 1964) where existence is shown under general conditions. Computing, however, the actual transformation is a completely different challenge with one of the possible solutions rely-1 Department of Computer Science, Rutgers University, New Brunswick, NJ, USA. 2 Department of Electrical and Computer Engineering, University of Patras, Patras, Greece.. Correspondence to: K. Basioti kib21@scarletmail.rutgers.edu