Many deep generative models are defined as a push-forward of a Gaussian measure by a continuous generator, such as Generative Adversarial Networks (GANs) or Variational Auto-Encoders (VAEs). This work explores the latent space of such deep generative models. A key issue with these models is their tendency to output samples outside of the support of the target distribution when learning disconnected distributions. We investigate the relationship between the performance of these models and the geometry of their latent space. Building on recent developments in geometric measure theory, we prove a sufficient condition for optimality in the case where the dimension of the latent space is larger than the number of modes. Through experiments on GANs, we demonstrate the validity of our theoretical results and gain new insights into the latent space geometry of these models. Additionally, we propose a truncation method that enforces a simplicial cluster structure in the latent space and improves the performance of GANs.

翻译：许多深度生成模型，如生成对抗网络（GANs）或变分自编码器（VAEs），被定义为通过连续生成器对高斯测度的推前映射。本研究探讨了此类深度生成模型的潜在空间。这些模型的一个关键问题是，在学习不连通的分布时，容易生成目标分布支撑集之外的样本。我们研究了模型性能与其潜在空间几何结构之间的关系。基于几何测度论的最新进展，我们证明了在潜在空间维度大于模态数量情况下的最优性充分条件。通过在GANs上的实验，我们验证了理论结果的有效性，并获得了对这些模型潜在空间几何结构的新见解。此外，我们提出了一种截断方法，该方法在潜在空间中强加单形簇结构，并提升了GANs的性能。