A.1 ProofofTheorem1 Proof log Ex g(x)[f(x) ] Ex g(x)[logf(x) ]=log

In order apply the change-of-variables formula to get a density for the generator, we assume that G: Rd RD spans an immersedd-dimensional manifold inRD. The governing assumption is that the Jacobian ofG exist and has full rank. However, we note that one requirement is that no hidden layer may have dimensionality below the d dimensions of the latent space. This is a natural requirement for the generator anyway. In our model, we aim to maximize the entropy of the generator, which encourages the generator to create as diverse samples as possible.