Data-sparse settings such as robotic manipulation, molecular physics, and galaxy morphology classification are some of the hardest domains for deep learning. For these problems, equivariant networks can help improve modeling across undersampled parts of the input space, and uncertainty estimation can guard against overconfidence. However, until now, the relationships between equivariance and model confidence, and more generally equivariance and model calibration, has yet to be studied. Since traditional classification and regression error terms show up in the definitions of calibration error, it is natural to suspect that previous work can be used to help understand the relationship between equivariance and calibration error. In this work, we present a theory relating equivariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors (ECE and ENCE) under various equivariance conditions, we elucidate the generalization limits of equivariant models and illustrate how symmetry mismatch can result in miscalibration in both classification and regression. We complement our theoretical framework with numerical experiments that clarify the relationship between equivariance and uncertainty using a variety of real and simulated datasets, and we comment on trends with symmetry mismatch, group size, and aleatoric and epistemic uncertainties.

翻译：在机器人操控、分子物理和星系形态分类等数据稀疏场景中，深度学习面临严峻挑战。对于此类问题，等变网络有助于改善输入空间欠采样区域的建模效果，而不确定性估计能够防范模型过度自信。然而，迄今为止，等变性与模型置信度之间的关系，以及更广义的等变性与模型校准之间的关联，尚未得到系统研究。由于传统分类与回归误差项直接体现在校准误差的定义中，我们有理由推测先前研究成果可用于解析等变性对校准误差的影响机制。本研究提出了一套关联等变性与不确定性估计的理论框架。通过证明不同等变条件下不确定性校准误差（ECE与ENCE）的上下界，我们揭示了等变模型的泛化极限，并阐明对称性失配如何导致分类与回归任务中的校准偏差。我们通过数值实验补充理论分析，利用真实与模拟数据集的多维度验证，厘清了等变性与不确定性的内在关联，并对对称性失配、群规模以及偶然性与认知性不确定性的影响趋势进行了系统性阐释。