Topological degree as a discrete diagnostic for disentanglement, with applications to the $\Delta$VAE

TL;DR

This paper introduces a topological diagnostic based on the degree of the encoder to evaluate disentanglement in Diffusion Variational Autoencoders with spherical latent spaces, demonstrating its effectiveness through theoretical and experimental analysis.

Contribution

It proposes a novel topological degree diagnostic for disentanglement, applying homology theory to analyze the encoder's structure in $ riangle$VAE models.

Findings

Encoder degree converges to ±1 after training, indicating a homotopy to a homeomorphism.

The diagnostic effectively assesses the topological properties of learned representations.

$ riangle$VAE achieves low LSBD scores, showing good disentanglement.

Abstract

We investigate the ability of Diffusion Variational Autoencoder ($\Delta$VAE) with unit sphere $\mathcal{S}^2$ as latent space to capture topological and geometrical structure and disentangle latent factors in datasets. For this, we introduce a new diagnostic of disentanglement: namely the topological degree of the encoder, which is a map from the data manifold to the latent space. By using tools from homology theory, we derive and implement an algorithm that computes this degree. We use the algorithm to compute the degree of the encoder of models that result from the training procedure. Our experimental results show that the $\Delta$VAE achieves relatively small LSBD scores, and that regardless of the degree after initialization, the degree of the encoder after training becomes $-1$ or $+1$, which implies that the resulting encoder is at least homotopic to a homeomorphism.