A Geometric View of Optimal Transportation and Generative Model

TL;DR

This paper explores the deep geometric connections between optimal transportation and convex geometry, proposing a novel framework for generative models that simplifies training by leveraging the Kantorovich potential.

Contribution

It introduces a geometric interpretation of generative models based on optimal transportation, enabling direct computation of the transportation map and simplifying the training process.

Findings

Geometric interpretation improves approximation of multi-cluster distributions.

Discriminator computes Kantorovich potential, generator computes transportation map.

Method outperforms WGAN in low-dimensional clustering tasks.

Abstract

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.