Wasserstein Auto-Encoders (1711.01558v4)

Abstract: We propose the Wasserstein Auto-Encoder (WAE)---a new algorithm for building a generative model of the data distribution. WAE minimizes a penalized form of the Wasserstein distance between the model distribution and the target distribution, which leads to a different regularizer than the one used by the Variational Auto-Encoder (VAE). This regularizer encourages the encoded training distribution to match the prior. We compare our algorithm with several other techniques and show that it is a generalization of adversarial auto-encoders (AAE). Our experiments show that WAE shares many of the properties of VAEs (stable training, encoder-decoder architecture, nice latent manifold structure) while generating samples of better quality, as measured by the FID score.

Overview of Wasserstein Auto-Encoders

The paper "Wasserstein Auto-Encoders" introduces a novel approach to generative modeling through the formulation of Wasserstein Auto-Encoders (WAE). This method aims to provide a balance between the strengths of Variational Auto-Encoders (VAE) and Generative Adversarial Networks (GAN), addressing the limitations inherent in each. The authors present a framework for minimizing the optimal transport distance between the generated and true data distributions, employing different regularization strategies to enforce this objective.

Key Contributions

The primary contributions of the paper can be summarized as follows:

1. New Family of Regularized Auto-Encoders:
   • The paper defines WAEs by minimizing the optimal transport cost $W_c(P_X, P_G)$ between the true data distribution $P_X$ and the model distribution $P_G$.
   • Two terms constitute the WAE objective: the reconstruction cost and a regularizer $\mathcal{D}_Z(P_Z, Q_Z)$, the latter penalizing the divergence between the prior distribution $P_Z$ and the encoded distribution $Q_Z$.
2. Theoretical Insights and Derivations:
   • The authors build on the theoretical framework of optimal transport. Specifically, they show the equivalence between the primal form of the optimal transport cost and a problem involving the optimization of a probabilistic encoder $Q(Z|X)$.
   • They provide a new perspective on the generative modeling problem by integrating optimal transport theory, which aligns with the weaker topological properties advantageous in high-dimensional settings.
3. Implementation of WAE with Different Regularizers:
   • The paper proposes two variants: WAE-GAN and WAE-MMD, based on adversarial training and Maximum Mean Discrepancy (MMD) respectively.
   • WAE-GAN employs a GAN-like setup in the latent space, using an adversary to match the encoded and prior distributions.
   • WAE-MMD uses MMD to compare these distributions, offering a fully adversary-free optimization scenario.
4. Empirical Evaluation:
   • The authors extensively evaluate WAEs on well-known datasets such as MNIST and CelebA.
   • Results indicate that WAEs maintain the advantageous properties of VAEs, such as stable training and good latent manifold structure, while also generating higher-quality samples, approaching the performance of GANs as measured by the Fréchet Inception Distance (FID) score.

Experimental Insights

The paper reports a comprehensive evaluation of the proposed WAE models. Key findings can be summarized as:

• Sample Quality:
  • WAE-GAN, in particular, produces high-quality samples that are competitive with GANs, highlighting the effectiveness of the adversarial regularizer.
  • WAE-MMD also shows considerable improvement over traditional VAEs, particularly in maintaining sample diversity and sharpness.
• Stability:
  • WAE-MMD proves to be more stable during training compared to WAE-GAN, thanks to its adversary-free nature.
  • Both WAE variants provide stable encoder-decoder architectures, maintaining the consistency in the latent space representation.
• Reconstruction Accuracy:
  • WAEs exhibit strong reconstruction capabilities, which is crucial for applications requiring accurate feature retrieval.
  • The method demonstrates robustness in interpolating between latent codes, further validating the smooth encoding-decoding process enabled by the objective formulation.

Practical and Theoretical Implications

The implications of WAEs are significant both in theory and practice:

• Theoretical Contributions:
  • The paper bridges the gap between optimal transport theory and practical generative modeling. By doing so, it provides a more grounded understanding of divergence minimization in variational approaches.
• Practical Applications:
  • The flexibility of WAEs, accommodating various regularizers, makes them suitable for a broad array of applications, from image generation to representation learning.
  • The improved sample quality and stable training dynamics presented by WAEs pave the way for more reliable and practical deployment of generative models in real-world scenarios.

Future Directions

The research opens several avenues for future exploration:

• Alternative Regularizers:
  • Exploring additional divergence measures or even adversarially trained cost functions in the input space $X$ could further enhance model performance.
• Dual Formulations:
  • A deeper theoretical analysis of the dual formulations for WAE-GAN and WAE-MMD can yield more insights into their properties and potential improvements.
• Scalability and Applications:
  • Evaluating the scalability of WAEs in higher-dimensional and more complex datasets can establish their robustness.
  • Investigating the practical aspects of WAEs in domains such as video synthesis, text generation, and other modalities remains an open and exciting area of research.

In conclusion, Wasserstein Auto-Encoders represent a notable advancement in the generative modeling landscape. By skillfully integrating optimal transport theory with practical auto-encoder designs, the authors provide both a theoretical framework and a practical tool that push the boundaries of latent variable models.