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Adaptive Learning of the Latent Space of Wasserstein Generative Adversarial Networks

Yixuan Qiu, Qingyi Gao, Xiao Wang

TL;DR

A novel framework called the latent Wasserstein GAN (LWGAN) is proposed that fuses the Wasserstein auto-encoder and the Wasserstein GAN so that the intrinsic dimension of the data manifold can be adaptively learned by a modified informative latent distribution.

Abstract

Generative models based on latent variables, such as generative adversarial networks (GANs) and variational auto-encoders (VAEs), have gained lots of interests due to their impressive performance in many fields. However, many data such as natural images usually do not populate the ambient Euclidean space but instead reside in a lower-dimensional manifold. Thus an inappropriate choice of the latent dimension fails to uncover the structure of the data, possibly resulting in mismatch of latent representations and poor generative qualities. Towards addressing these problems, we propose a novel framework called the latent Wasserstein GAN (LWGAN) that fuses the Wasserstein auto-encoder and the Wasserstein GAN so that the intrinsic dimension of the data manifold can be adaptively learned by a modified informative latent distribution. We prove that there exist an encoder network and a generator network in such a way that the intrinsic dimension of the learned encoding distribution is equal to the dimension of the data manifold. We theoretically establish that our estimated intrinsic dimension is a consistent estimate of the true dimension of the data manifold. Meanwhile, we provide an upper bound on the generalization error of LWGAN, implying that we force the synthetic data distribution to be similar to the real data distribution from a population perspective. Comprehensive empirical experiments verify our framework and show that LWGAN is able to identify the correct intrinsic dimension under several scenarios, and simultaneously generate high-quality synthetic data by sampling from the learned latent distribution.

Adaptive Learning of the Latent Space of Wasserstein Generative Adversarial Networks

TL;DR

A novel framework called the latent Wasserstein GAN (LWGAN) is proposed that fuses the Wasserstein auto-encoder and the Wasserstein GAN so that the intrinsic dimension of the data manifold can be adaptively learned by a modified informative latent distribution.

Abstract

Generative models based on latent variables, such as generative adversarial networks (GANs) and variational auto-encoders (VAEs), have gained lots of interests due to their impressive performance in many fields. However, many data such as natural images usually do not populate the ambient Euclidean space but instead reside in a lower-dimensional manifold. Thus an inappropriate choice of the latent dimension fails to uncover the structure of the data, possibly resulting in mismatch of latent representations and poor generative qualities. Towards addressing these problems, we propose a novel framework called the latent Wasserstein GAN (LWGAN) that fuses the Wasserstein auto-encoder and the Wasserstein GAN so that the intrinsic dimension of the data manifold can be adaptively learned by a modified informative latent distribution. We prove that there exist an encoder network and a generator network in such a way that the intrinsic dimension of the learned encoding distribution is equal to the dimension of the data manifold. We theoretically establish that our estimated intrinsic dimension is a consistent estimate of the true dimension of the data manifold. Meanwhile, we provide an upper bound on the generalization error of LWGAN, implying that we force the synthetic data distribution to be similar to the real data distribution from a population perspective. Comprehensive empirical experiments verify our framework and show that LWGAN is able to identify the correct intrinsic dimension under several scenarios, and simultaneously generate high-quality synthetic data by sampling from the learned latent distribution.
Paper Structure (30 sections, 8 theorems, 92 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 30 sections, 8 theorems, 92 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. GAN and WGAN.
  3. VAE and WAE.
  4. Issues of Latent Dimension Mismatch
  5. The Latent Wasserstein GAN
  6. Existence of optimal encoder-generator pairs
  7. The proposed model
  8. Computational algorithm
  9. Tuning parameter selection
  10. Theoretical Results
  11. Generalization error bound
  12. Estimation consistency
  13. Intrinsic dimension consistency
  14. Experimental Results
  15. Simulated experiments
  16. ...and 15 more sections

Key Result

Theorem 1

If $d\ge r$, then there exist two continuous mappings $Q^{\diamond}:\mathcal{X}\rightarrow\mathcal{Z}$ and $G^{\diamond}:\mathcal{Z}\rightarrow\mathcal{X}$ such that $Q^{\diamond}(X)\sim N(0,A_{r})$ and $X=G^{\diamond}(Q^{\diamond}(X))$.

Figures (9)

  • Figure 1: Illustrations of data generation with wrong latent dimensions in WGAN and WAE.
  • Figure 2: Simulated data supported on manifolds and the demonstrations of the fitted LWAGN models.
  • Figure 3: Digits 1 (top row) and 2 (bottom row) of the MNIST data, and the demonstrations of the fitted LWAGN models.
  • Figure 4: MNIST data with all digits and the demonstrations of the fitted LWAGN model.
  • Figure 5: True sample of the preprocessed CelebA dataset and the rank score plot to estimate the intrinsic dimension.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Definition 1: Topological manifold, lee2013introduction
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Theorem 3
  • Theorem 4
  • ...and 5 more