Mode Collapse and Regularity of Optimal Transportation Maps

Abstract

This work builds the connection between the regularity theory of optimal transportation map, Monge-Ampère equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. According to the regularity theory of Monge-Ampère equation, if the support of the target measure is disconnected or just non-convex, the optimal transportation mapping is discontinuous. General DNNs can only approximate continuous mappings. This intrinsic conflict leads to the convergence difficulty and mode collapse in GANs. We test our hypothesis that the supports of real data distribution are in general non-convex, therefore the discontinuity is unavoidable using an Autoencoder combined with discrete optimal transportation map (AE-OT framework) on the CelebA data set. The testing result is positive. Furthermore, we propose to approximate the continuous Brenier potential directly based on discrete Brenier theory to tackle mode collapse. Comparing with existing method, this method is more accurate and effective.

References in corpus (15)

• Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks
• Generative Adversarial Text to Image Synthesis
• Progressive Growing of GANs for Improved Quality, Stability, and Variation
• Improved Training of Wasserstein GANs
• Improved Techniques for Training GANs
• NIPS 2016 Tutorial: Generative Adversarial Networks
• Which Training Methods for GANs do actually Converge?
• Adversarial Feature Learning
• Adversarially Learned Inference
• VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning
• Mode Regularized Generative Adversarial Networks
• Amortised MAP Inference for Image Super-resolution
• Do GANs actually learn the distribution? An empirical study
• On the Limitations of First-Order Approximation in GAN Dynamics
• Relaxed Wasserstein with Applications to GANs

Cited by in corpus (5)

• Optimal transport mapping via input convex neural networks
• 2-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANs
• LPIPS-AttnWav2Lip: Generic Audio-Driven lip synchronization for Talking Head Generation in the Wild
• Global regularity for the Monge-Ampère equation with natural boundary condition
• Convex Smoothed Autoencoder-Optimal Transport model