Quantifying Manifolds: Do the manifolds learned by Generative Adversarial Networks converge to the real data manifold

TL;DR

The paper addresses how GANs learn data manifolds and whether these learned manifolds converge to the real data manifold. It adopts Topological Data Analysis (TDA) by leveraging persistence diagrams and Vietoris–Rips filtrations to quantify intrinsic dimensions and topological features ($H_0$, $H_1$, $H_2$) of both real and generated data across training epochs, using entropy and Wasserstein distances as comparators. Experiments on CIFAR-10 cat images show that the GAN-generated manifold's intrinsic dimension and topological summaries tend to converge toward the real data manifold over training, with $H_0$ converging faster than higher homology groups. The work presents a multi-metric framework for GAN manifold quantification, offering a principled lens on robustness and data geometry and suggesting broader applicability to other generative models.

Abstract

This paper presents our experiments to quantify the manifolds learned by ML models (in our experiment, we use a GAN model) as they train. We compare the manifolds learned at each epoch to the real manifolds representing the real data. To quantify a manifold, we study the intrinsic dimensions and topological features of the manifold learned by the ML model, how these metrics change as we continue to train the model, and whether these metrics convergence over the course of training to the metrics of the real data manifold.

Figures (2)

• Figure 1: Intrinsic Dimension vs Epoch
• Figure 2: Topological Features vs Epoch

Theorems & Definitions (4)

• definition 1
• definition 2
• definition 3: Manifold
• definition 4