Monotone Generative Modeling via a Gromov-Monge Embedding
Wonjun Lee Yifei Yang Dongmian Zou Gilad Lerman
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.