The presence of symmetries imposes a stringent set of constraints on a system. This constrained structure allows intelligent agents interacting with such a system to drastically improve the efficiency of learning and generalization, through the internalisation of the system's symmetries into their information-processing. In parallel, principled models of complexity-constrained learning and behaviour make increasing use of information-theoretic methods. Here, we wish to marry these two perspectives and understand whether and in which form the information-theoretic lens can "see" the effect of symmetries of a system. For this purpose, we propose a novel variant of the Information Bottleneck principle, which has served as a productive basis for many principled studies of learning and information-constrained adaptive behaviour. We show (in the discrete case) that our approach formalises a certain duality between symmetry and information parsimony: namely, channel equivariances can be characterised by the optimal mutual information-preserving joint compression of the channel's input and output. This information-theoretic treatment furthermore suggests a principled notion of "soft" equivariance, whose "coarseness" is measured by the amount of input-output mutual information preserved by the corresponding optimal compression. This new notion offers a bridge between the field of bounded rationality and the study of symmetries in neural representations. The framework may also allow (exact and soft) equivariances to be automatically discovered.

翻译：对称性的存在对系统施加了一组严格的约束。这种受约束的结构使得与此类系统交互的智能体能够通过将系统的对称性内化到其信息处理过程中，显著提升学习与泛化的效率。与此同时，基于复杂性约束的学习与行为的理论模型越来越多地采用信息论方法。在此，我们希望融合这两种视角，探究信息论视角是否以及以何种形式能够“看见”系统对称性的影响。为此，我们提出了一种基于信息瓶颈原则的新变体，该原则一直是许多关于学习及信息约束下适应性行为的理论研究的富有成效的基础。我们证明（在离散情况下）我们的方法形式化了对称性与信息简约性之间的某种对偶性：即通道等变性可以通过对通道输入和输出进行最优互信息保持的联合压缩来刻画。这种信息论处理方式还提出了一种原则性的“软”等变性概念，其“粗糙度”由相应最优压缩所保留的输入-输出互信息量来衡量。这一新概念为有限理性领域与神经表征中对称性的研究之间架起了桥梁。该框架还可能使（精确和软）等变性得以自动发现。