Disentangled representation learning is a challenging task that involves separating multiple factors of variation in complex data. Although various metrics for learning and evaluating disentangled representations have been proposed, it remains unclear what these metrics truly quantify and how to compare them. In this work, we study the definitions of disentanglement given by first-order equational predicates and introduce a systematic approach for transforming an equational definition into a compatible quantitative metric based on enriched category theory. Specifically, we show how to replace (i) equality with metric or divergence, (ii) logical connectives with order operations, (iii) universal quantifier with aggregation, and (iv) existential quantifier with the best approximation. Using this approach, we derive metrics for measuring the desired properties of a disentangled representation extractor and demonstrate their effectiveness on synthetic data. Our proposed approach provides practical guidance for researchers in selecting appropriate evaluation metrics and designing effective learning algorithms for disentangled representation learning.

本文研究了解离散表示学习的定义，并介绍了将等式定义转化为基于丰富范畴论的相容量化评估标准的系统方法。我们提出了适用于评估去迭代复杂数据的可分离表示抽取器所需性质的量化度量标准，并展示了它们在合成数据上的有效性。我们的方法为研究人员选择合适的评估标准和设计有效的离散表示学习算法提供了实用指导。

丰富分离：从定义到度量