In recent years there has been increased interest in understanding the interplay between deep generative models (DGMs) and the manifold hypothesis. Research in this area focuses on understanding the reasons why commonly-used DGMs succeed or fail at learning distributions supported on unknown low-dimensional manifolds, as well as developing new models explicitly designed to account for manifold-supported data. This manifold lens provides both clarity as to why some DGMs (e.g. diffusion models and some generative adversarial networks) empirically surpass others (e.g. likelihood-based models such as variational autoencoders, normalizing flows, or energy-based models) at sample generation, and guidance for devising more performant DGMs. We carry out the first survey of DGMs viewed through this lens, making two novel contributions along the way. First, we formally establish that numerical instability of high-dimensional likelihoods is unavoidable when modelling low-dimensional data. We then show that DGMs on learned representations of autoencoders can be interpreted as approximately minimizing Wasserstein distance: this result, which applies to latent diffusion models, helps justify their outstanding empirical results. The manifold lens provides a rich perspective from which to understand DGMs, which we aim to make more accessible and widespread.

深度生成模型与流形假设之间的相互作用引起了人们的广泛关注。本文通过流形视角对DGM进行了首次调查，并对其进行了两个新的贡献：首先，形式上证明了高维似然函数的数值不稳定性是无法避免的；其次，发现基于自编码器的DGM可以被解释为近似最小化Wasserstein距离，这一结果有助于解释其出色的实证结果。流形视角提供了理解DGM的丰富视角，我们希望能够使之更加易于理解和推广。

通过流形假设的视角解析深度生成模型：一项调查与新连接