Incorporating geometric inductive biases into models can aid interpretability and generalization, but encoding to a specific geometric structure can be challenging due to the imposed topological constraints. In this paper, we theoretically and empirically characterize obstructions to training encoders with geometric latent spaces. We show that local optima can arise due to singularities (e.g. self-intersection) or due to an incorrect degree or winding number. We then discuss how normalizing flows can potentially circumvent these obstructions by defining multimodal variational distributions. Inspired by this observation, we propose a new flow-based model that maps data points to multimodal distributions over geometric spaces and empirically evaluate our model on 2 domains. We observe improved stability during training and a higher chance of converging to a homeomorphic encoder.

通过将几何归纳偏差引入模型可以提高其解释性和泛化性，但由于拓扑约束的存在，将其编码到特定的几何结构中可能很具有挑战性。本文从理论和实证研究角度对训练编码器时遇到的几何潜空间障碍进行了特征化。我们展示了由于奇异性（例如自交）或不正确的度数或缠数可能导致局部最优解的情况。然后讨论了如何通过定义多模式变分分布，潜在地规避这些障碍的可能性。受此观察的启发，我们提出了一种新的基于流的模型，将数据点映射到几何空间上的多模式分布，并在两个领域上进行了实证评估。我们观察到训练过程中的稳定性提高，出现同胚编码器的概率增加。

拓扑障碍及其规避方法