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许多任务中被证明有效，但其成功高度依赖于一个假设：真实函数在整个定义域内对称性，与等变神经网络中的对称性相匹配。在等变学习文献中，一个缺失的部分是当对称性仅在定义域部分存在时，对等变网络的分析。在本工作中，我们提出了针对此类情况的通用理论。我们提出了正确、错误和外部等变性的逐点定义，从而能够连续量化函数所展现的每种等变性的程度。随后，我们研究了不同程度的错误或外部对称性对模型误差的影响。我们证明，在具有部分错误对称性的分类或回归设定中，不变网络或等变神经网络存在误差下界。我们还分析了外部等变性可能带来的有害影响。实验在三种不同环境中验证了这些结果。