VAEs and GANs: Implicitly Approximating Complex Distributions with Simple Base Distributions and Deep Neural Networks -- Principles, Necessity, and Limitations (2503.01898v1)

Abstract: This tutorial focuses on the fundamental architectures of Variational Autoencoders (VAE) and Generative Adversarial Networks (GAN), disregarding their numerous variations, to highlight their core principles. Both VAE and GAN utilize simple distributions, such as Gaussians, as a basis and leverage the powerful nonlinear transformation capabilities of neural networks to approximate arbitrarily complex distributions. The theoretical basis lies in that a linear combination of multiple Gaussians can almost approximate any probability distribution, while neural networks enable further refinement through nonlinear transformations. Both methods approximate complex data distributions implicitly. This implicit approximation is crucial because directly modeling high-dimensional distributions explicitly is often intractable. However, the choice of a simple latent prior, while computationally convenient, introduces limitations. In VAEs, the fixed Gaussian prior forces the posterior distribution to align with it, potentially leading to loss of information and reduced expressiveness. This restriction affects both the interpretability of the model and the quality of generated samples.

Overview of Implicit Approximation in VAEs and GANs

The paper "VAEs and GANs: Implicitly Approximating Complex Distributions with Simple Base Distributions and Deep Neural Networks - Principles, Necessity, and Limitations" offers an in-depth analysis of the foundational principles underlying Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs). By focusing exclusively on vanilla architectures, the paper elucidates the fundamental mechanism both models employ: approximating complex data distributions via simple latent distributions and nonlinear transformations facilitated by neural networks.

The central premise of the paper is that direct modeling of high-dimensional distributions in an explicit manner is often computationally infeasible. VAEs and GANs circumvent this by sampling from simplified distributions, such as Gaussians, and applying nonlinear transformations through neural networks to produce outputs that approach the target distribution. Such an implicit approximation leverages the mathematical property that linear combinations of simple distributions can approximate more complex ones, which is further refined through neural network transformations.

Core Principles and Implications

The paper begins by exploring the architecture and objective functions of VAEs and GANs. In VAEs, the intrinsic limitation arises from the fixed Gaussian prior used for the latent space, which necessitates alignment with the posterior distribution via the Kullback-Leibler (KL) divergence in the objective function. This constraint can result in diminished model expressiveness and potentially lower quality in the generated samples, often characterized by blurred outputs. The GAN architecture, involving a discriminator and generator, similarly relies on a standard normal distribution to guide the generation of complex structures.

Despite these limitations, the implicit approximation technique is deemed necessary for modeling high-dimensional data. The paper exemplifies this necessity through the intractability of explicitly formulating high-dimensional data distributions, such as those needed for realistic image generation.

Advantages and Limitations of Simple Latent Priors

The use of simple distributions, like the standard normal distribution, as latent priors offers computational efficiency and streamlined sampling processes. However, the choice introduces significant trade-offs. The predetermined nature of the prior in VAEs can constrain their inference capability and interpretability, impacting both the expressed posterior distribution and the resultant generation quality. This fixed simple prior limits the model's capacity to capture the true underlying latent distribution and therefore impacts the capacity to derive meaningful, interpretable relationships within the latent space.

The research further discusses potential avenues for improvement. By adopting adaptive latent priors, VAEs and GANs could potentially enhance their inferential insights and generative capabilities. The paper thus hints at the interplay between inference and generation, positing that improved inference mechanisms might lead to more robust generative outcomes.

Conclusion

The paper provides a comprehensive overview of how VAEs and GANs implicitly approximate complex distributions via simple base distributions and neural networks. While outlining the necessity and rationality of the implicit approach for high-dimensional data, it also critically evaluates the implications of using simple latent priors. By highlighting these trade-offs, the paper offers a foundational perspective on the limitations and potential boundaries within which future generative model advancements could operate, such as incorporating more sophisticated priors to enhance model interpretability and output quality.

This analysis contributes to the ongoing discourse on improving generative models, emphasizing the need to balance computational feasibility with the pursuit of enhancing inference and generative precision. Further research could explore the integration of more adaptive mechanisms to address the identified limitations, thereby pushing the boundaries of current generative architecture capabilities within AI.