Enriching Disentanglement: Definitions to Metrics

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.