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）定义为高斯测度在连续生成器下的推前测度。本研究探索此类深度生成模型的潜空间。这些模型的一个关键问题在于：当学习不连通分布时，它们倾向于生成目标分布支撑集以外的样本。我们研究了模型性能与潜空间几何结构之间的关系。基于几何测度论的最新进展，我们证明了在潜空间维度大于模态数量情况下的最优性充分条件。通过GANs实验，我们验证了理论结果的有效性，并对这些模型的潜空间几何结构获得了新的认识。此外，我们提出了一种截断方法，该方法在潜空间中强制构建单纯复形簇结构，从而提升了GANs的性能。