Although equivariant machine learning has proven effective at many tasks, success depends heavily on the assumption that the ground truth function is symmetric over the entire domain matching the symmetry in an equivariant neural network. A missing piece in the equivariant learning literature is the analysis of equivariant networks when symmetry exists only partially in the domain. In this work, we present a general theory for such a situation. We propose pointwise definitions of correct, incorrect, and extrinsic equivariance, which allow us to quantify continuously the degree of each type of equivariance a function displays. We then study the impact of various degrees of incorrect or extrinsic symmetry on model error. We prove error lower bounds for invariant or equivariant networks in classification or regression settings with partially incorrect symmetry. We also analyze the potentially harmful effects of extrinsic equivariance. Experiments validate these results in three different environments.

翻译：尽管等变机器学习在众多任务中已证明其有效性，但其成功很大程度上依赖于一个假设：真实函数在整个定义域上的对称性与等变神经网络中的对称性相匹配。等变学习文献中缺失的一环，是对称性仅部分存在于定义域时等变网络的分析。本文针对此类情况提出通用理论。我们提出正确等变性、错误等变性与外部等变性的逐点定义，从而连续量化函数呈现的每类等变性的程度。继而研究不同程度的错误或外部对称性对模型误差的影响。在分类或回归场景中，针对具有部分错误对称性的不变或等变网络，我们证明了误差下界。同时分析外部等变性的潜在危害。实验在三种不同环境中验证了这些结论。