Towards Information Theory-Based Discovery of Equivariances

• URL: http://arxiv.org/abs/2310.16555v4
• Date: Wed, 02 Oct 2024 10:43:23 GMT
• Title: Towards Information Theory-Based Discovery of Equivariances
• Authors: Hippolyte Charvin, Nicola Catenacci Volpi, Daniel Polani,
• Abstract summary: The presence of symmetries imposes a stringent set of constraints on a system. In parallel, principled models of complexity-constrained learning and behaviour make increasing use of information-theoretic methods. 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.
• Score: 1.2289361708127875
• License:
• Abstract: 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 and under a specific technical assumption) 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.

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