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.

通过使用 Gromov-Monge 嵌入（GME）的深度生成模型，我们提出一种解决生成对抗网络（GANs）面临的起始条件敏感性和模式崩溃等挑战的方法。我们在低维的潜在空间中通过 GME 识别数据的底层测度的低维结构，并在保持其几何性的同时将其映射到一个测度，然后将其最优地转运到参考测度。我们通过 GME 的几何性和生成映射的 $c$- 循环单调性来保持底层几何的一致性，这是确保更好的参数初始化和模式崩溃鲁棒性的第一步。数值实验证明了我们方法在生成高质量图像、避免模式崩溃以及对不同的起始条件具有鲁棒性方面的有效性。