  Unsupervised domain mapping has attracted substantial attention in recent
years due to the success of models based on the cycle-consistency assumption.
These models map between two domains by fooling a probabilistic discriminator,
thereby matching the probability distributions of the real and generated data.
Instead of this probabilistic approach, we cast the problem in terms of
aligning the geometry of the manifolds of the two domains. We introduce the
Manifold Geometry Matching Generative Adversarial Network (MGM GAN), which adds
two novel mechanisms to facilitate GANs sampling from the geometry of the
manifold rather than the density and then aligning two manifold geometries: (1)
an importance sampling technique that reweights points based on their density
on the manifold, making the discriminator only able to discern geometry and (2)
a penalty adapted from traditional manifold alignment literature that
explicitly enforces the geometry to be preserved. The MGM GAN leverages the
manifolds arising from a pre-trained autoencoder to bridge the gap between
formal manifold alignment literature and existing GAN work, and demonstrate the
advantages of modeling the manifold geometry over its density.
