Limits of Generalized Magnitudes of Dissimilarity Spaces

Abstract

For a finite symmetric nonnegative dissimilarity matrix $D=(d_{ij})$ with zero diagonal, we study the high-resolution limit ($t\to \infty$) of $\mathbf{1}^T(e^{-td_{ij}})^+\mathbf{1}$. We prove that this limit always exists in the extended real line and is rational whenever finite, without weightability or invertibility assumptions. At fixed cardinality, a finite rational polyhedral fan governs rank and leading asymptotics. Consequently, only finitely many finite limits occur, and every limit admits an integer-data realization at the same cardinality. In contrast, allowing cardinality to vary, every rational pair of zero-pattern magnitude and high-resolution limit is realized using dissimilarities in $\{0,1,2\}$; adding the level $3$ permits either infinite limit with any rational zero-pattern magnitude. We also classify all limits on at most four points. Moreover, we derive an exact Schur decomposition valid under persistent singularity. Finally, it is shown that positive semidefiniteness at some positive scale is equivalent to equality of dissimilarity rows for every zero-distance pair, in which case the limit counts the zero-distance classes.