正确、错误与外在等变性的通用理论
A General Theory of Correct, Incorrect, and Extrinsic Equivariance
AI总结：
本文针对真实对称性仅在部分定义域存在的情形，提出正确、错误与外在等变性的逐点定义和通用理论，证明部分错误对称性下不变/等变网络的误差下界，并分析外在等变性的潜在危害，实验验证了理论结果。
摘要：
尽管等变机器学习已在许多任务中被证明有效，但其成功在很大程度上依赖于一个假设：真实函数在整个定义域上具有与等变神经网络相匹配的对称性。等变学习文献中缺失的一环是，当对称性仅在定义域中部分存在时对等变网络的分析。本文针对这种情况提出了一套通用理论。我们提出了正确等变性、错误等变性和外在等变性的逐点定义，从而能够连续地量化一个函数所表现出的各类等变性程度。随后，我们研究了不同程度的错误或外在对称性对模型误差的影响。我们证明了在分类或回归设置中，具有部分错误对称性的不变或等变网络的误差下界。我们还分析了外在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。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
