Unveiling the Latent Space Geometry of Push-Forward Generative Models

TL;DR

This paper investigates the geometric structure of the latent space in deep generative models like GANs and VAEs, providing theoretical insights and a new truncation method to improve sample quality and model performance.

Contribution

It offers a geometric measure theory-based condition for optimality in latent spaces and introduces a truncation technique that enhances GAN performance.

Findings

Theoretical condition for optimality in high-dimensional latent spaces.

Validation of theoretical results through GAN experiments.

A novel truncation method that enforces a simplicial structure, improving model outputs.

Abstract

Many deep generative models are defined as a push-forward of a Gaussian measure by a continuous generator, such as Generative Adversarial Networks (GANs) or Variational Auto-Encoders (VAEs). This work explores the latent space of such deep generative models. A key issue with these models is their tendency to output samples outside of the support of the target distribution when learning disconnected distributions. We investigate the relationship between the performance of these models and the geometry of their latent space. Building on recent developments in geometric measure theory, we prove a sufficient condition for optimality in the case where the dimension of the latent space is larger than the number of modes. Through experiments on GANs, we demonstrate the validity of our theoretical results and gain new insights into the latent space geometry of these models. Additionally, we propose a truncation method that enforces a simplicial cluster structure in the latent space and improves the performance of GANs.