Transformation Categorization Based on Group Decomposition Theory Using Parameter Division

A team led by Takayuki Komatsu at the University of Tokyo has introduced a novel unsupervised method for categorizing transformations between pairs of inputs, grounded in group decomposition theory. Their approach, called parameter division, splits a single transformation's parameter into components and applies homomorphism constraints mapping the full transformation to one component. This allows them to identify the normal subgroup—the set of transformations where that component is fixed to the identity—thereby categorizing transformations like rotation, translation, and scale without requiring auxiliary assumptions such as motion or isometry restrictions that plagued prior Galois-theoretic methods.

In experiments on image pairs, the method successfully learned distinct categories for rotation, translation, and scale, with ablation studies confirming that the group-decomposition constraints—not auxiliary assumptions—drive the categorization. This work addresses a fundamental gap in representation learning: classical disentanglement assumes mutually independent factors, failing when they are coupled. By leveraging algebraic constraints, the approach offers a principled, theoretically grounded alternative for discovering meaningful sensory representations, with potential applications in robotics, computer vision, and unsupervised learning systems that need to understand how objects transform without explicit labels.