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.

本研究解决了Diffusion Variational Autoencoder ($\Delta$VAE)在捕捉数据集的拓扑和几何结构及解缠潜在因子方面的能力不足的问题。我们提出了一种新的解缠诊断方法——编码器的拓扑度，通过同调理论的工具开发并实施了计算算法。实验结果表明，$\Delta$VAE模型的编码器在训练后其拓扑度变为 $-1$ 或 $+1$，暗示该编码器至少是同胚于同构。

拓扑度作为解缠的离散诊断，及其在 $Δ$VAE 中的应用