# Transformation Categorization Based on Group Decomposition Theory Using Parameter Division

arXiv: https://arxiv.org/abs/2605.04056 · HTML: https://mlsanity.com/paper/2605.04056/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 14 references that are in the map.
- First posted on arXiv 10 Apr 2026.

## Builds on (most cited first, 7 of 7)

- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Disentangling Patterns and Transformations from One Sequence of Images with Shape-invariant Lie Group Transformer](https://mlsanity.com/paper/2203.11210/) (2022)
- [Unsupervised categorization of similarity measures](https://mlsanity.com/paper/2502.08098/) (2025)
- [Learning Conditionally Independent Transformations using Normal Subgroups in Group Theory](https://mlsanity.com/paper/2504.04490/) (2025)
- [Feature-Based Lie Group Transformer for Real-World Applications](https://mlsanity.com/paper/2506.04668/) (2025)

## Built on by (most cited first, 1 of 1)

- [Unsupervised Learning of Inter-Object Relationships via Group Homomorphism](https://mlsanity.com/paper/2604.20925/) (2026)

## Abstract (arXiv)

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.