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Analyzing and Improving Optimal-Transport-based Adversarial Networks

Jaemoo Choi, Jaewoong Choi, Myungjoo Kang

TL;DR

This paper unifies these OT-based adversarial methods within a single framework, elucidate the role of each component in training dynamics through a comprehensive analysis of this unified framework, and suggests a simple but novel method that improves the previously best-performing OT-based model.

Abstract

Optimal Transport (OT) problem aims to find a transport plan that bridges two distributions while minimizing a given cost function. OT theory has been widely utilized in generative modeling. In the beginning, OT distance has been used as a measure for assessing the distance between data and generated distributions. Recently, OT transport map between data and prior distributions has been utilized as a generative model. These OT-based generative models share a similar adversarial training objective. In this paper, we begin by unifying these OT-based adversarial methods within a single framework. Then, we elucidate the role of each component in training dynamics through a comprehensive analysis of this unified framework. Moreover, we suggest a simple but novel method that improves the previously best-performing OT-based model. Intuitively, our approach conducts a gradual refinement of the generated distribution, progressively aligning it with the data distribution. Our approach achieves a FID score of 2.51 on CIFAR-10 and 5.99 on CelebA-HQ-256, outperforming unified OT-based adversarial approaches.

Analyzing and Improving Optimal-Transport-based Adversarial Networks

TL;DR

This paper unifies these OT-based adversarial methods within a single framework, elucidate the role of each component in training dynamics through a comprehensive analysis of this unified framework, and suggests a simple but novel method that improves the previously best-performing OT-based model.

Abstract

Optimal Transport (OT) problem aims to find a transport plan that bridges two distributions while minimizing a given cost function. OT theory has been widely utilized in generative modeling. In the beginning, OT distance has been used as a measure for assessing the distance between data and generated distributions. Recently, OT transport map between data and prior distributions has been utilized as a generative model. These OT-based generative models share a similar adversarial training objective. In this paper, we begin by unifying these OT-based adversarial methods within a single framework. Then, we elucidate the role of each component in training dynamics through a comprehensive analysis of this unified framework. Moreover, we suggest a simple but novel method that improves the previously best-performing OT-based model. Intuitively, our approach conducts a gradual refinement of the generated distribution, progressively aligning it with the data distribution. Our approach achieves a FID score of 2.51 on CIFAR-10 and 5.99 on CelebA-HQ-256, outperforming unified OT-based adversarial approaches.
Paper Structure (44 sections, 4 theorems, 25 equations, 19 figures, 9 tables, 1 algorithm)

This paper contains 44 sections, 4 theorems, 25 equations, 19 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Background and Related Works
  4. Kantorovich OT
  5. OT Loss in GANs
  6. OT Map as Generative model
  7. Analyzing OT-based Adversarial Approaches
  8. A Unified Framework
  9. Comparative Analysis of OT-based GANs
  10. Experimental Settings
  11. Effect of Strictly Convex $g_1$ and $g_2$
  12. Experimental Results
  13. Effect of $g_1$ and $g_2$ in Optimization
  14. Effect of Cost Function
  15. Experimental Results
  16. ...and 29 more sections

Key Result

Theorem 3.1

Let $g_1$ and $g_2$ be real-valued functions that are non-decreasing, bounded below, differentiable, and strictly convex. Assuming the regularity assumptions in Appendix appen:proofs, there exists a unique Lipschitz continuous optimal potential $v^\star$ for Eq. eq:semi-dual-uot. Moreover, for the s is equi-bounded and equi-Lipschitz.

Figures (19)

  • Figure 1: Comparison of Training Dynamics between OT-based GANs.Left: Visualization of generated samples (blue) and data samples (red) for every 6K iterations. Right: Training loss of the generator ($T_\theta$ Loss) and discriminator ($v_\phi$ Loss) for each algorithm.
  • Figure 2: Quantitative Evaluation of OT-based GANs on CIFAR-10.
  • Figure 3: Image Generation on CIFAR-10.$\dagger$ indicates the results conducted by ourselves.
  • Figure 4: Qualitative Comparison of Generated Samples from UOTM.Left: When $\tau$ is too small ($\tau=0.0002$). Middle: When $\tau$ is optimal ($\tau=0.001$). Right: When $\tau$ is too large ($\tau=0.005$). On Left, we reordered randomly generated samples to gather similar-looking samples. When $\tau$ is large, the samples appear noisy, and when $\tau$ is small, the generated samples show a mode collapse problem.
  • Figure 5: Visualization of Generator $T_{\theta}$. The gray lines illustrate the generated pairs, i.e., the connecting lines between $x$ (green) and $T_{\theta}(x)$ (blue). The red dots represent the training data samples.
  • ...and 14 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 4.1
  • Theorem A.1
  • proof
  • Theorem A.2
  • proof