A Differentiable Metric for Discovering Groups and Unitary Representations
Dongsung Huh
Discovering group structures in data poses a fundamental challenge across diverse scientific domains. The primary obstacle lies in the non-differentiable nature of group axioms, impeding their integration into deep learning framework. To overcome this, we present a novel differentiable approach leveraging the representation theory of finite groups. Our method features 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. Crucially, our model implicitly defines a complexity metric that favors the discovery of group structures. Evaluations demonstrate our method's ability to accurately recover group operations and learn their unitary representations from partial observation. Our work lays the foundation for a promising new paradigm in automated algebraic structure discovery, with far-reaching applications across diverse domains, particularly in enabling automatic symmetry discovery for geometric deep learning.
group theory, representation theory, representation learning, symmetry discovery, symbolic relationship