VAE与GAN用简单分布+神经网络隐式逼近复杂数据分布
VAEs and GANs: Implicitly Approximating Complex Distributions with Simple Base Distributions and Deep Neural Networks -- Principles, Necessity, and Limitations
- 用高斯等简单分布作基础，通过神经网络非线性变换逼近复杂分布
- 隐式逼近避免了高维分布显式建模的计算困境
- 固定先验限制表达能力，影响生成质量与模型可解释性
本教程聚焦变分自编码器（VAE）和生成对抗网络（GAN）的核心架构，忽略其众多变体，以阐明其基本原理。两者均采用高斯等简单分布作为基础，并利用神经网络强大的非线性变换能力，逼近任意复杂的概率分布。理论依据在于：多个高斯分布的线性组合几乎可逼近任意概率分布，而神经网络则通过非线性变换实现进一步优化。两种方法均以隐式方式逼近复杂数据分布，这一机制至关重要，因为直接显式建模高维分布通常不可行。然而，选择简单潜在先验虽具计算优势，也带来局限性：在VAE中，固定的高斯先验迫使后验分布与其对齐，可能导致信息损失与表达能力下降，进而影响模型可解释性与生成样本质量。
原文摘要 · Abstract (English)
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.
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