Learning Conditionally Independent Transformations using Normal Subgroups in Group Theory
This addresses a fundamental limitation in unsupervised learning for computer vision, though it appears incremental as it extends existing algebraic approaches to noncommutative cases.
The paper tackles the challenge of separating conditionally independent but noncommutative transformations in unsupervised representation learning by proposing a method based on normal subgroups from group theory. Experiments on geometric image transformations show successful categorization of transformations like rotation and translation.
Humans develop certain cognitive abilities to recognize objects and their transformations without explicit supervision, highlighting the importance of unsupervised representation learning. A fundamental challenge in unsupervised representation learning is to separate different transformations in learned feature representations. Although algebraic approaches have been explored, a comprehensive theoretical framework remains underdeveloped. Existing methods decompose transformations based on algebraic independence, but these methods primarily focus on commutative transformations and do not extend to cases where transformations are conditionally independent but noncommutative. 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 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 that leverages normal subgroups to enable the separation of conditionally independent transformations, even in the absence of commutativity. Through experiments on geometric transformations in images, we show that our method successfully categorizes conditionally independent transformations, such as rotation and translation, in an unsupervised manner, suggesting a close link between group decomposition via normal subgroups and transformation categorization in representation learning.