# Generating and Aligning from Data Geometries with Generative Adversarial Networks

## TL;DR

This paper introduces MGM GAN, a novel approach that aligns data geometries using manifold geometry matching, offering an alternative to probabilistic distribution matching in unsupervised domain mapping.

## Contribution

It proposes a new method combining importance sampling and geometry-preserving penalties to align data manifolds, bridging manifold alignment and GAN techniques.

## Key findings

- MGM GAN effectively aligns data geometries without density matching.
- The method improves domain translation quality in experiments.
- It leverages pre-trained autoencoders for manifold extraction.

## Abstract

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.