Generative Adversarial Networks (GANs) are powerful tools for creating new content, but they face challenges such as sensitivity to starting conditions and mode collapse. To address these issues, we propose a deep generative model that utilizes the Gromov-Monge embedding (GME). It helps identify the low-dimensional structure of the underlying measure of the data and then maps it, while preserving its geometry, into a measure in a low-dimensional latent space, which is then optimally transported to the reference measure. We guarantee the preservation of the underlying geometry by the GME and $c$-cyclical monotonicity of the generative map, where $c$ is an intrinsic embedding cost employed by the GME. The latter property is a first step in guaranteeing better robustness to initialization of parameters and mode collapse. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images, avoiding mode collapse, and exhibiting robustness to different starting conditions.

翻译：生成对抗网络（GANs）是创建新内容的强大工具，但面临对初始条件敏感和模式坍塌等挑战。为解决这些问题，我们提出一种深度生成模型，该模型利用Gromov-Monge嵌入（GME）。该嵌入有助于识别数据底层测度的低维结构，并在保持几何结构的同时将其映射到低维潜空间中的测度，随后通过最优传输将该测度映射到参考测度。我们确保GME对底层几何结构的保持以及生成映射的$c$-循环单调性，其中$c$是GME采用的内禀嵌入代价。后一性质是保证对参数初始化和模式坍塌具有更强鲁棒性的第一步。数值实验证明了我们的方法在生成高质量图像、避免模式坍塌以及展现对不同初始条件鲁棒性方面的有效性。