  In this work, we show the intrinsic relations between optimal transportation
and convex geometry, especially the variational approach to solve Alexandrov
problem: constructing a convex polytope with prescribed face normals and
volumes. This leads to a geometric interpretation to generative models, and
leads to a novel framework for generative models. By using the optimal
transportation view of GAN model, we show that the discriminator computes the
Kantorovich potential, the generator calculates the transportation map. For a
large class of transportation costs, the Kantorovich potential can give the
optimal transportation map by a close-form formula. Therefore, it is sufficient
to solely optimize the discriminator. This shows the adversarial competition
can be avoided, and the computational architecture can be simplified.
Preliminary experimental results show the geometric method outperforms WGAN for
approximating probability measures with multiple clusters in low dimensional
space.
