Atlas flow : compatible local structures on the manifold

TL;DR

Atlas flow introduces a topologically consistent generative model that leverages local manifold structures and compatibility conditions to improve understanding and generation of complex data manifolds, including StyleGAN2's style space.

Contribution

The paper proposes Atlas flow, a novel flow-based model that enforces local-to-global manifold compatibility, enhancing data generation and analysis of complex data manifolds.

Findings

Effective on synthetic manifold datasets with noise

Successfully models StyleGAN2's style vector manifold

Ensures topological consistency in local-global manifold integration

Abstract

In this paper, we focus on the intersections of a manifold's local structures to analyze the global structure of a manifold. We obtain local regions on data manifolds such as the latent space of StyleGAN2, using Mapper, a tool from topological data analysis. We impose gluing compatibility conditions on overlapping local regions, which guarantee that the local structures can be glued together to the global structure of a manifold. We propose a novel generative flow model called Atlas flow that uses compatibility to reattach the local regions. Our model shows that the generating processes perform well on synthetic dataset samples of well-known manifolds with noise. Furthermore, we investigate the style vector manifold of StyleGAN2 using our model.