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.

Ablation Study to Clarify the Mechanism of Object Segmentation in Multi-Object Representation Learning

Multi-object representation learning aims to represent complex real-world visual input using the composition of multiple objects. Representation learning methods have often used unsupervised learning to segment an input image into individual objects and encode these objects into each latent vector. However, it is not clear how previous methods have achieved the appropriate segmentation of individual objects. Additionally, most of the previous methods regularize the latent vectors using a Variational Autoencoder (VAE). Therefore, it is not clear whether VAE regularization contributes to appropriate object segmentation. To elucidate the mechanism of object segmentation in multi-object representation learning, we conducted an ablation study on MONet, which is a typical method. MONet represents multiple objects using pairs that consist of an attention mask and the latent vector corresponding to the attention mask. Each latent vector is encoded from the input image and attention mask. Then, the component image and attention mask are decoded from each latent vector. The loss function of MONet consists of 1) the sum of reconstruction losses between the input image and decoded component image, 2) the VAE regularization loss of the latent vector, and 3) the reconstruction loss of the attention mask to explicitly encode shape information. We conducted an ablation study on these three loss functions to investigate the effect on segmentation performance. Our results showed that the VAE regularization loss did not affect segmentation performance and the others losses did affect it. Based on this result, we hypothesize that it is important to maximize the attention mask of the image region best represented by a single latent vector corresponding to the attention mask. We confirmed this hypothesis by evaluating a new loss function with the same mechanism as the hypothesis.