Chart Auto-Encoders for Manifold Structured Data

Stephan Schonsheck, Jie Chen, Rongjie Lai

Keywords: Auto-encoder, differential manifolds, multi-charted latent space

TL;DR: Manifold-structured latent space for generative models

Abstract: Auto-encoding and generative models have made tremendous successes in image and signal representation learning and generation. These models, however, generally employ the full Euclidean space or a bounded subset (such as $[0,1]^l$) as the latent space, whose trivial geometry is often too simplistic to meaningfully reflect the structure of the data. This paper aims at exploring a nontrivial geometric structure of the latent space for better data representation. Inspired by differential geometry, we propose \textbf{Chart Auto-Encoder (CAE)}, which captures the manifold structure of the data with multiple charts and transition functions among them. CAE translates the mathematical definition of manifold through parameterizing the entire data set as a collection of overlapping charts, creating local latent representations. These representations are an enhancement of the single-charted latent space commonly employed in auto-encoding models, as they reflect the intrinsic structure of the manifold. Therefore, CAE achieves a more accurate approximation of data and generates realistic new ones. We conduct experiments with synthetic and real-life data to demonstrate the effectiveness of the proposed CAE.

Code: https://anonymous.4open.science/r/a40668ab-7542-4028-8709-694142a985da/