A Transport-Information Framework for Continuous-Fiber-Aligned Coarse-Graining Representation Sets

Abstract

Consider representation sets of interest whose elements are associated with continuous realizations related to them through coarse-graining (CG). CG can be regarded as an operation that reduces a representation by discarding information that is not retained in its output. However, a systematic description of the information structure associated with CG in general remains to be established. Here, we propose a transport-information framework for continuous-fiber-aligned representation sets, in which each representation is associated with a continuous realization through a fiber structure, with CG defining the projection onto the representation. Within this framework, CG is described through a local-map operation that transports continuous probability mass onto representative points. The resulting transport cost is expressed as the contraction of the second-moment structure of the underlying continuous representation with the transport metric. Building on the previously proposed framework of unavoidable canonical nonlinearity (UCN), this construction provides a non-arbitrary information-theoretic interpretation of the CG-associated transport cost in terms of information indistinguishability, quantified by an expected Kullback-Leibler divergence associated with the CG geometry. The present framework provides a basis for describing information costs associated with individual CG operations and for comparing representation sets while accounting for their underlying continuous realizations.