Transformation Categorization Based on Group Decomposition Theory Using Parameter Division

Takayuki Komatsu; Yoshiyuki Ohmura; Yasuo Kuniyoshi

cs.LGApr 10, 2026

Transformation Categorization Based on Group Decomposition Theory Using Parameter Division

Authors: Takayuki Komatsu, Yoshiyuki Ohmura, Yasuo Kuniyoshi

Organizations: Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan. · Next Generation Artificial Intelligence Research Center (AI Center), The University of Tokyo, Tokyo, Japan.

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.

Figures & tables

Fig. 1 : Schematic of the proposed formulation in the case of two parameters. By the closure property of a group

G

, when this group contains

g_{1}=g(\theta_{1},\phi_{1})

and

g_{2}=g(\theta_{2},\phi_{2})

, there exists

g_{1,2}=g(\theta_{1,2},\phi_{1,2})

which satisfies

g_{1,2}=g_{1}\circ g_{2}

g_{1}

g_{2}

, and

g_{1,2}

are mapped to

\theta_{1}

\theta_{2}

, and

\theta_{1,2}

, respectively, by the mapping

f\colon g(\theta,\phi)\mapsto\theta

. If

f

is a homomorphism,

\theta_{1,2}=\theta_{1}\cdot\theta_{2}

should hold. Using this homomorphism

f

, we can obtain the normal subgroup

N

G

as the set of

g

under the condition that

\theta

is set to the identity element

\theta_{e}

Panel of Figure 2 — Fig. 2 : Boxplots comparing the evaluation metric $I_{h}$ with and without the homomorphism loss $\mathcal{L}_{h}$ . Boxplot (a) is for the dataset variant (i) rotation and translation, (b) is for the dataset variant (ii) scale and rotation, and (c) is for the dataset variant (iii) scale and translation.

Panel of Figure 3 — Fig. 3 : Examples of the learning results.

Unsupervised Representation Learning

Unsupervised representation learning discovers useful, structured features from data without relying on task labels.

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Disentangled Representation Learning through Unsupervised Symmetry Group Discovery

Barthélémy Dang-Nhu, Louis Annabi, Sylvain Argentieri

Sorbonne Universit´e, CNRS, Institut des Syst`emes Intelligents et de Robotique, ISIR, F-75005 Paris, France

Disentangled Representation Learning Unsupervised Representation Learning Symmetry Discovery Equivariant Representation Learning

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Symmetry-based disentangled representation learning leverages the group structure of environment transformations to uncover the latent factors of variation. Prior approaches to symmetry-based disentanglement have required strong prior knowledge of the symmetry group's structure, or restrictive assumptions about the subgroup properties. In this work, we remove these constraints by proposing a method whereby an embodied agent autonomously discovers the group structure of its action space through unsupervised interaction with the environment. We prove the identifiability of the true symmetry group decomposition under minimal assumptions, and derive two algorithms: one for discovering the group decomposition from interaction data, and another for learning Linear Symmetry-Based Disentangled (LSBD) representations without assuming specific subgroup properties. Our method is validated on three environments exhibiting different group decompositions, where it outperforms existing LSBD approaches.

Abstract

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