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系列严格的约束。这种受约束的结构使得与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，大幅提升学习与泛化的效率。与此同时，针对复杂度受限的学习与行为所建立的原理性模型，正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，探究信息论视角能否“洞察”系统对称性的影响及其具体形式。为此，我们提出了一种信息瓶颈原理的新变体，该原理此前已为许多关于学习与信息受限自适应行为的原理性研究提供了富有成效的基础。我们证明（在离散情形下），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关系：即通道等变性可以通过保持最优互信息的前提下对通道输入与输出进行联合压缩来刻画。此外，这种信息论处理方法还提出了一种关于“软”等变性的原理性概念，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。这一新概念在有限理性研究与神经表征中的对称性研究之间架起了一座桥梁。该框架还可能使（精确与软）等变性实现自动发现。