  Generative adversarial networks (GANs) are popular for generative tasks;
however, they often require careful architecture selection, extensive empirical
tuning, and are prone to mode collapse. To overcome these challenges, we
propose a novel model that identifies the low-dimensional structure of the
underlying data distribution, maps it into a low-dimensional latent space while
preserving the underlying geometry, and then optimally transports a reference
measure to the embedded distribution. We prove three key properties of our
method: 1) The encoder preserves the geometry of the underlying data; 2) The
generator is $c$-cyclically monotone, where $c$ is an intrinsic embedding cost
employed by the encoder; and 3) The discriminator's modulus of continuity
improves with the geometric preservation of the data. Numerical experiments
demonstrate the effectiveness of our approach in generating high-quality images
and exhibiting robustness to both mode collapse and training instability.
