We develop a new, unsupervised symmetry learning method that starts with raw data, and gives the minimal (discrete) generator of an underlying Lie group of symmetries, together with a symmetry equivariant representation of the data. The method is able to learn the pixel translation operator from a dataset with only an approximate translation symmetry, and can learn quite different types of symmetries which are not apparent to the naked eye, equally well. The method is based on the formulation of an information-theoretic loss function that measures both the degree to which the dataset is symmetric under a given candidate symmetry, and also, the degree of locality of the samples in the dataset with respect to this symmetry. We demonstrate that this coupling between symmetry and locality, together with a special optimization technique developed for entropy estimation, results in a highly stable system that gives reproducible results. The symmetry actions we consider are group representations, however, we believe the approach has the potential to be generalized to more general, nonlinear actions of non-commutative Lie groups.

翻译：我们提出了一种新的无监督对称性学习方法，该方法从原始数据出发，能够给出底层李群对称性的最小（离散）生成元，并同时生成数据的对称等变表示。该方法能够从仅具有近似平移对称性的数据集中学习像素平移算子，并且同样能够有效学习肉眼难以察觉的多种不同类型的对称性。该方法基于信息论损失函数的构建，该函数同时度量了数据集在给定候选对称性下的对称程度，以及数据集样本相对于该对称性的局部性程度。我们证明，这种对称性与局部性的耦合，结合为熵估计专门开发的优化技术，形成了一个高度稳定的系统，能够产生可复现的结果。我们所考虑的对称作用是群表示，但我们相信该方法有潜力推广到更一般的非交换李群非线性作用。