Relational Compression: A Framework for Relational Fidelity in Constrained Representations

What should a compressed representation preserve when the information of interest lies in relationships among elements rather than in the elements themselves? We formulate relational compression in the classical source-description-reconstruction sense, but with relational structure itself as the fidelity-bearing content. Each instance specifies the source relation, retained description, reconstructed or evaluated relation, fidelity criterion, and constrained resource. We use this interface to situate selected methods from graph summarization, spectral sparsification, similarity-preserving representation, and relational distillation within a common formulation while keeping their different reconstruction and resource assumptions explicit. We develop finite-codeword collision as one concrete realization. Same-codeword probability yields a relational geometry linking pair-specific alignment and separation to aggregate R'enyi-2 occupancy and the spherical geometry of categorical assignments, with exact objective correspondences to squared-Euclidean centroid reconstruction and normalized graph association and cut. Graph and image studies illustrate complementary routes within the finite-codeword family: graph- and teacher-defined relational requirements act directly on equality or collision, while reconstruction acts through a joint decoder. Together, these results illustrate how distinct relational requirements can be formulated and tested within a common constrained-representation framework.