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.

翻译：解耦表示学习是一项具有挑战性的任务，涉及分离复杂数据中的多个变化因素。尽管已有多种用于学习和评估解耦表示的度量方法被提出，但这些度量究竟量化了什么以及如何进行比较仍不明确。在本研究中，我们探讨了基于一阶等式谓词的解耦定义，并引入了一种系统化方法，基于丰富范畴论将等式定义转化为兼容的定量度量。具体而言，我们展示了如何将（i）等式替换为度量或散度，（ii）逻辑连接词替换为序运算，（iii）全称量词替换为聚合操作，以及（iv）存在量词替换为最佳近似。利用该方法，我们推导出了用于度量解耦表示提取器所需属性的指标，并在合成数据上验证了其有效性。我们提出的方法为研究人员在选择适当评估指标和设计高效解耦表示学习算法方面提供了实用指导。