Monotone Generative Modeling via a Gromov-Monge Embedding
This work addresses generative modeling for AI/ML researchers by offering a more stable and robust alternative to GANs, though it appears incremental as it builds on existing optimal transport and geometric embedding concepts.
The paper tackles the challenges of GANs, such as mode collapse and instability, by proposing a model that identifies low-dimensional data structure, preserves geometry in a latent space, and uses optimal transport for generation, resulting in high-quality image generation with improved robustness.
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.