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.
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 g1=g(θ1,ϕ1) and g2=g(θ2,ϕ2) , there exists g1,2=g(θ1,2,ϕ1,2) which satisfies g1,2=g1∘g2 . g1 , g2 , and g1,2 are mapped to θ1 , θ2 , and θ1,2 , respectively, by the mapping f:g(θ,ϕ)↦θ . If f is a homomorphism, θ1,2=θ1⋅θ2 should hold. Using this homomorphism f , we can obtain the normal subgroup N of G as the set of g under the condition that θ is set to the identity element θe .
Fig. 2 : Boxplots comparing the evaluation metric Ih with and without the homomorphism loss Lh . 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.
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.
Barthélémy Dang-Nhu, Louis Annabi, Sylvain Argentieri
Sorbonne Universit´e, CNRS, Institut des Syst`emes Intelligents et de Robotique, ISIR, F-75005 Paris, France
Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict g1⋆g2 for elements of a finite group G. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.
Jianliang He, Leda Wang, Fengzhuo Zhang +2
Department of Statistics and Data Science, Yale University
While current deep learning models achieve high performance by learning statistical correlations from vast datasets,which stands in stark contrast to human learning. They lack the flexibility of humans-particularly preverbal infants-to autonomously acquire the underlying structure of the world from limited experience and adapt to novel situations. In this study, we propose an unsupervised representation learning method based on a hierarchical relationship in group operations, rather than statistical independence, aiming to build a computational model of the cognitive development of infants. The proposed model features an integrated architecture that simultaneously performs object segmentation and the extraction of motion laws from dynamic image sequences. By introducing the Homomorphism from algebra as a structural constraint within a neural network, the model structurally separates pixel-level changes into meaningful, decomposed transformation components, such as translation and deformation. Using interaction scenes (chasing and evading tasks) based on developmental science findings, we experimentally demonstrate that the model can segment multiple objects into individual slots without any ground-truth labels. Furthermore, we confirmed that relative movements between objects, such as approaching or receding, are accurately mapped and structured into a one-dimensional additive latent space. These results suggest that by introducing algebraic geometric constraints rather than relying solely on statistical correlation learning, physically interpretable "disentangled representations" can be acquired. This study contributes to the understanding of the process by which infants internalize environmental laws as structures and provides a new perspective for constructing artificial systems with developmental intelligence.