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.

HyperCube网络引入了一种新的方法来自主发现数据中的对称群结构，该方法利用了唯一因子化架构和新颖的正则化器，以朝向学习正交表示的有力归纳偏置，并成功地从部分观察到的数据中高效地学习了一般的群操作，同时恢复了完整的操作表，这些因子直接对应于底层群的精确矩阵表示，而且这些因子捕捉到了群的完整不可约表示集合，形成执行群卷积的广义傅里叶基础。大量与群和非群符号操作的实验显示，与Transformer基准相比，HyperCube在训练速度上表现出了显著的100-1000倍改进和2-10倍的更高样本效率，这些结果表明我们的方法开启了一类能够利用数据内在对称性的深度学习模型，从而在性能和广泛适用性上取得了显著的改进。

通过隐式正交性偏好发现对称群结构