  Variational Auto-Encoders enforce their learned intermediate latent-space
data distribution to be a simple distribution, such as an isotropic Gaussian.
However, this causes the posterior collapse problem and loses manifold
structure which can be important for datasets such as facial images. A GAN can
transform a simple distribution to a latent-space data distribution and thus
preserve the manifold structure, but optimizing a GAN involves solving a
Min-Max optimization problem, which is difficult and not well understood so
far. Therefore, we propose a GAN-like method to transform a simple distribution
to a data distribution in the latent space by solving only a minimization
problem. This minimization problem comes from training a discriminator between
a simple distribution and a latent-space data distribution. Then, we can
explicitly formulate an Optimal Transport (OT) problem that computes the
desired mapping between the two distributions. This means that we can transform
a distribution without solving the difficult Min-Max optimization problem.
Experimental results on an eight-Gaussian dataset show that the proposed OT can
handle multi-cluster distributions. Results on the MNIST and the CelebA
datasets validate the effectiveness of the proposed method.
