A General Theory of Correct, Incorrect, and Extrinsic Equivariance
This work addresses a foundational gap in equivariant learning for researchers, providing theoretical insights into partial symmetry scenarios, though it is incremental in extending existing equivariance theory.
The paper tackles the problem of equivariant machine learning when symmetry only partially exists in the domain, proposing definitions to quantify correct, incorrect, and extrinsic equivariance and analyzing their impact on model error. It proves error lower bounds for networks with partially incorrect symmetry and validates results in three experimental environments.
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.