A Differentiable Metric for Discovering Groups and Unitary Representations

ICLR 2025 Conference Submission13021 Authors

Keywords: group theory, representation theory, representation learning, symmetry discovery, symbolic relationship

Abstract: Discovering group structures within data is a significant challenge with broad implications across various scientific domains. The main hurdle stems from the non-differentiable nature of group axioms, hindering their seamless integration into deep learning frameworks. To address this, we introduce a novel differentiable approach that leverages the representation theory of finite groups. Our method employs a unique neural network architecture that models interactions between group elements as multiplications of their matrix representations, coupled with a regularizer that promotes unitarity of these matrices. Furthermore, our model implicitly defines a complexity metric that prioritizes the discovery of group structures. In numerical evaluation, our method successfully recovers group operations from a limited number of observations as well as accurately learning their unitary representations. This work establishes a new avenue for uncovering groups within data, with potential applications in diverse fields, including automatic symmetry discovery in deep learning.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 13021