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

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.