Transformation Categorization Based on Group Decomposition Theory Using Parameter Division

TL;DR

This paper introduces a novel parameter division method for unsupervised transformation categorization, leveraging group decomposition theory to improve representation learning without auxiliary assumptions.

Contribution

It proposes a new parameter division approach that generalizes group decomposition for transformation categorization, removing previous auxiliary constraints.

Findings

The method effectively categorizes transformations like rotation, translation, and scale.

Ablation studies confirm the importance of group-decomposition constraints.

The approach broadens the applicability of algebraic transformation analysis.

Abstract

Representation learning seeks meaningful sensory representations without supervision and can model aspects of human development. Although many neural networks empirically learn useful features, a principled account of what makes a representation "good" remains elusive. We study unsupervised categorization of transformations between pairs of inputs under algebraic constraints. Classical disentanglement favors mutually independent factors and fails when factors are coupled. Our prior Galois-theoretic approach decomposes a group via normal subgroups by learning a product of two transformations with one factor constrained to a normal subgroup, covering both commutative and non-commutative cases. That method, however, relied on auxiliary assumptions (e.g., motion and isometry restrictions) not required by decomposition theory, and ablations did not separate theory-based from auxiliary effects. We propose parameter division for a single transformation: we split its parameter into components, impose homomorphism constraints mapping the full transformation to one component, and identify the normal subgroup as the set of transformations when that component is fixed to the identity. This formulation drops the previous auxiliary assumptions and applies more broadly. We evaluate on image pairs involving rotation, translation, and scale; ablations show that group-decomposition constraints drive appropriate categorization.