Disentangling the explanatory factors in complex data is a promising approach for generalizable and data-efficient representation learning. While a variety of quantitative metrics for learning and evaluating disentangled representations have been proposed, it remains unclear what properties these metrics truly quantify. In this work, we establish a theoretical connection between logical definitions of disentanglement and quantitative metrics using topos theory and enriched category theory. We introduce a systematic approach for converting a first-order predicate into a real-valued quantity by replacing (i) equality with a strict premetric, (ii) the Heyting algebra of binary truth values with a quantale of continuous values, and (iii) quantifiers with aggregators. The metrics induced by logical definitions have strong theoretical guarantees, and some of them are easily differentiable and can be used as learning objectives directly. Finally, we empirically demonstrate the effectiveness of the proposed metrics by isolating different aspects of disentangled representations.

翻译：将复杂数据中的解释性因素解耦，是实现泛化能力强且数据高效的表示学习的一种有前景的方法。尽管已有多种用于学习和评估解耦表示的量化指标被提出，但这些指标究竟量化了哪些属性仍不明确。在本工作中，我们利用拓扑斯理论和丰富范畴论，在解耦的逻辑定义与量化指标之间建立了理论联系。我们引入了一种系统方法，将一阶谓词转化为实数值：通过（i）将等式替换为严格预度量，（ii）将二元真值的海廷代数替换为连续值的量子格，以及（iii）将量词替换为聚合器。由逻辑定义导出的指标具有强大的理论保障，其中一些易于微分，可直接用作学习目标。最后，我们通过分离解耦表示的不同方面，实验证明了所提指标的有效性。