A Category-theoretical Meta-analysis of Definitions of Disentanglement
Yivan Zhang, Masashi Sugiyama
TL;DR
This work tackles the fragmentation of disentanglement definitions by offering a category-theoretic unification centered on cartesian and monoidal products. It shows how modularity (product morphisms) and explicitness (split monomorphisms) characterize disentangled representations across sets, relations, and stochastic maps, using frameworks like functor categories and Markov categories. By reinterpreting generating, encoding, and decoding processes within these structures, the paper connects equivariant maps, probabilistic dependencies, and structured kernels under a common formalism. While theoretical, this perspective provides a principled basis for selecting definitions and motivates future work to derive metrics and scalable, task-informed criteria.
Abstract
Disentangling the factors of variation in data is a fundamental concept in machine learning and has been studied in various ways by different researchers, leading to a multitude of definitions. Despite the numerous empirical studies, more theoretical research is needed to fully understand the defining properties of disentanglement and how different definitions relate to each other. This paper presents a meta-analysis of existing definitions of disentanglement, using category theory as a unifying and rigorous framework. We propose that the concepts of the cartesian and monoidal products should serve as the core of disentanglement. With these core concepts, we show the similarities and crucial differences in dealing with (i) functions, (ii) equivariant maps, (iii) relations, and (iv) stochastic maps. Overall, our meta-analysis deepens our understanding of disentanglement and its various formulations and can help researchers navigate different definitions and choose the most appropriate one for their specific context.