Feature-Based Lie Group Transformer for Real-World Applications

The main goal of representation learning is to acquire meaningful representations from real-world sensory inputs without supervision. Representation learning explains some aspects of human development. Various neural network (NN) models have been proposed that acquire empirically good representations. However, the formulation of a good representation has not been established. We recently proposed a method for categorizing changes between a pair of sensory inputs. A unique feature of this approach is that transformations between two sensory inputs are learned to satisfy algebraic structural constraints. Conventional representation learning often assumes that disentangled independent feature axes is a good representation; however, we found that such a representation cannot account for conditional independence. To overcome this problem, we proposed a new method using group decomposition in Galois algebra theory. Although this method is promising for defining a more general representation, it assumes pixel-to-pixel translation without feature extraction, and can only process low-resolution images with no background, which prevents real-world application. In this study, we provide a simple method to apply our group decomposition theory to a more realistic scenario by combining feature extraction and object segmentation. We replace pixel translation with feature translation and formulate object segmentation as grouping features under the same transformation. We validated the proposed method on a practical dataset containing both real-world object and background. We believe that our model will lead to a better understanding of human development of object recognition in the real world.

Learning Conditionally Independent Transformations using Normal Subgroups in Group Theory

Humans develop certain cognitive abilities to recognize objects and their transformations without explicit supervision, highlighting the importance of unsupervised representation learning. A fundamental challenge in unsupervised representation learning is to separate different transformations in learned feature representations. Although algebraic approaches have been explored, a comprehensive theoretical framework remains underdeveloped. Existing methods decompose transformations based on algebraic independence, but these methods primarily focus on commutative transformations and do not extend to cases where transformations are conditionally independent but noncommutative. To extend current representation learning frameworks, we draw inspiration from Galois theory, where the decomposition of groups through normal subgroups provides an approach for the analysis of structured transformations. Normal subgroups naturally extend commutativity under certain conditions and offer a foundation for the categorization of transformations, even when they do not commute. In this paper, we propose a novel approach that leverages normal subgroups to enable the separation of conditionally independent transformations, even in the absence of commutativity. Through experiments on geometric transformations in images, we show that our method successfully categorizes conditionally independent transformations, such as rotation and translation, in an unsupervised manner, suggesting a close link between group decomposition via normal subgroups and transformation categorization in representation learning.