We consider the problem of learning a function respecting a symmetry from among a class of symmetries. We develop a unified framework that enables symmetry discovery across a broad range of subgroups including locally symmetric, dihedral and cyclic subgroups. At the core of the framework is a novel architecture composed of linear, matrix-valued and non-linear functions that expresses functions invariant to these subgroups in a principled manner. The structure of the architecture enables us to leverage multi-armed bandit algorithms and gradient descent to efficiently optimize over the linear and the non-linear functions, respectively, and to infer the symmetry that is ultimately learnt. We also discuss the necessity of the matrix-valued functions in the architecture. Experiments on image-digit sum and polynomial regression tasks demonstrate the effectiveness of our approach.

翻译：我们考虑从一类对称性中学习满足对称性的函数的问题。我们开发了一个统一框架，能够在包括局部对称、二面体群和循环群在内的广泛子群类别中实现对称性发现。该框架的核心是一种由线性函数、矩阵值函数和非线性函数组成的新型架构，该架构能够以原则性的方式表达对这些子群具有不变性的函数。该架构的结构使得我们能够利用多臂老虎机算法和梯度下降分别高效地优化线性函数和非线性函数，并推断最终学习到的对称性。我们还讨论了该架构中矩阵值函数的必要性。在图像数字求和和多项式回归任务上的实验证明了我们方法的有效性。