Learning Conditionally Independent Transformations using Normal Subgroups in Group Theory

Humans develop certain cognitive abilities that enable them to recognize objects and their transformations without explicit supervision. This suggests the importance of developing models for unsupervised representation learning. In unsupervised representation learning, a fundamental challenge is to separate different transformations in learned feature representations. Although algebraic approaches have been explored to achieve this separation, a comprehensive theoretical framework remains underdeveloped. To address this, researchers have proposed methods that decompose independent transformations based on algebraic independence, which generalizes the mathematical notion of independence. However, these methods primarily focus on transformations that commute, which means that the order in which two transformations are applied does not affect the final result. The methods do not naturally extend to cases where transformations are conditionally independent but noncommutative, where changing the order of application leads to different outcomes. To extend current representation learning frameworks, we draw inspiration from Galois theory, where the decomposition of groups through normal subgroups provides an approach for the analysis of structured transformations. Normal subgroups naturally extend the concept of commutativity under certain conditions and offer a foundation for the categorization of transformations, even when they do not commute. In this paper, we propose a novel approach to representation learning that extends the theoretical framework of transformation decomposition by leveraging normal subgroups from Galois theory. Our method enables the separation of transformations that are conditionally independent, even in the absence of commutativity. In the experiment, we apply our method to the learning of geometric transformations in images and demonstrate that it successfully categorizes conditionally independent transformations, such as rotation and translation, in an unsupervised manner. This result suggests that the decomposition of groups via normal subgroups, as established in Galois theory, is closely related to the categorization of transformations in representation learning.