Optimal Transport for Generative Models

Optimal transport plays a fundamental role in deep learning. Natural data sets have intrinsic patterns, which can be summarized as the manifold distribution principle: a natural class of data can be treated as a probability distribution on a low-dimensional manifold, embedded in a high-dimensional ambient space. A deep learning system mainly accomplishes two tasks: manifold learning and probability distribution learning. Given a manifold X, all the probability measures on X form an infinite dimensional manifold, the so-called Wasserstein space. Optimal transport assigns a Riemannian metric on the Wasserstein space, the so-called Wasserstein metric, and defines Otto’s calculus, such that variational optimization can be carried out in the Wasserstein space P ( X ) $$\mathcal {P}(X)$$ . A deep learning system learns the distribution by optimizing some functionals in the Wasserstein space P ( X ) $$\mathcal {P}(X)$$ ; therefore optimal transport lays down the theoretic foundation for deep learning. This work introduces the theory of optimal transport and the profound relation between Brenier’s theorem and Alexandrov’s theorem in differential geometry via Monge-Ampère equation. We give a variational proof for Alexandrov’s theorem and convert the proof to a computational algorithm to solve the optimal transport maps. The algorithm is based on computational geometry and can be generalized to general manifold setting. Optimal transport theory and algorithms have been extensively applied in the models of generative adversarial networks (GANs). In a GAN model, the generator computes the optimal transport map (OT map), while the discriminator computes the Wasserstein distance between the generated data distribution and the real data distribution. The optimal transport theory shows the competition between the generator and the discriminator is completely unnecessary and should be replaced by collaboration. Furthermore, the regularity theory of optimal transport map explains the intrinsic reason for mode collapsing. A novel generative model is introduced, which uses an autoencoder (AE) for manifold learning and OT map for probability distribution transformation. This AE-OT model improves the theoretical rigor and transparency, as well as the computational stability and efficiency; in particular, it eliminates the mode collapsing.