用拓扑度诊断变分自编码器的解耦能力，发现训练后编码器接近双射。
Topological degree as a discrete diagnostic for disentanglement, with applications to the $Δ$VAE
- 引入编码器的拓扑度作为解耦新指标，基于同调论计算。
- ΔVAE训练后编码器拓扑度恒为±1，表明其近似双射。
- 适合研究表征学习与几何结构建模的学者参考。
我们研究了以单位球面 $\mathcal{S}^2$ 为潜在空间的扩散变分自编码器（ΔVAE）在捕捉数据集的拓扑与几何结构、实现潜在因子解耦方面的能力。为此，提出一种新的解耦诊断方法：即编码器的拓扑度，该编码器是从数据流形到潜在空间的映射。通过同调论工具，推导并实现了一种计算该度数的算法。利用该算法计算训练后模型编码器的拓扑度。实验结果表明，ΔVAE取得相对较低的 LSBD 分数；无论初始化时的拓扑度如何，训练后的编码器拓扑度均趋于 -1 或 +1，这表明最终编码器至少同伦于同胚映射。
原文摘要 · Abstract (English)
We investigate the ability of Diffusion Variational Autoencoder ($Δ$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 $Δ$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.
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