A Geometric View of Optimal Transportation and Generative Model
This work addresses the computational complexity and instability in GANs for machine learning practitioners, offering a more efficient approach, though it appears incremental as it builds on existing optimal transportation and GAN frameworks.
The paper tackles the problem of simplifying generative adversarial networks (GANs) by linking optimal transportation to convex geometry, showing that adversarial competition can be avoided and the architecture simplified, with preliminary results indicating the method outperforms WGAN in approximating multi-cluster probability measures in low-dimensional spaces.
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.