Discovering Symmetry Group Structures via Implicit Orthogonality Bias

We introduce the HyperCube network, a novel approach for autonomously discovering symmetry group structures within data. The key innovation is a unique factorization architecture coupled with a novel regularizer that instills a powerful inductive bias towards learning orthogonal representations. This leverages a fundamental theorem of representation theory that all compact/finite groups can be represented by orthogonal matrices. HyperCube efficiently learns general group operations from partially observed data, successfully recovering complete operation tables. Remarkably, the learned factors correspond directly to exact matrix representations of the underlying group. Moreover, these factors capture the group's complete set of irreducible representations, forming the generalized Fourier basis for performing group convolutions. In extensive experiments with both group and non-group symbolic operations, HyperCube demonstrates a dramatic 100-1000x improvement in training speed and 2-10x greater sample efficiency compared to the Transformer baseline. These results suggest that our approach unlocks a new class of deep learning models capable of harnessing inherent symmetries within data, leading to significant improvements in performance and broader applicability.