Abstract:

Generative Adversarial Learning (GAL) aims to learn a target distribution in an adversarial manner. A Generative Adversarial Network (GAN) is a concrete implementation of GAL using a discriminator and a generator that play a min-max game. GANs have been used in many machine learning and computer vision applications. However, GANs are known to be hard to train, mainly because a min-max saddle point optimization problem needs to be solved in GAL. In this thesis, I investigate several methods to improve generative adversarial learning using Optimal Transport (OT).

Previous Wasserstein GANs (WGANs) do not compute the correct Wasserstein distance to train the discriminator. To address this problem, I propose WGAN-TS that uses the L1 transport cost and computes the correct Wasserstein distance to train the discriminator. To ensure the local convergence of WGANs, I propose WGAN-QC that adopts the quadratic transport cost. I prove that WGAN-QC not only computes the correct Wasserstein distance but also converges to a local equilibrium point. To compute the Wasserstein distance over the whole dataset, I propose to use Semi-Discrete Optimal Transport (SDOT) to match noise points and the real images during GAN training. To measure the quality of an SDOT map, I use the Maximum Relative Error (MRE) and the L_1 distance between the target distribution and the transported distribution obtained by an OT map. I propose statistical methods to estimate the MRE and the L_1 distance. I propose an efficient Epoch Gradient Descent algorithm for SDOT (SDOT-EGD). To deal with the 2D special case of GAL, I propose to use OT to learn 2D distributions. In particular, I adopt OT to match persistent diagrams in training a topology-aware GAN and learn density maps in the crowd counting task. Finally, I use OT and the topological maps of the crowd to improve the crowd counting performance and propose a topology-based metric to measure the quality of the crowd density maps.