Convergence of Magnitude of Finite Positive Definite Metric Spaces
Byungchang So
TL;DR
The article investigates the continuity of magnitude under the Gromov-Hausdorff distance for finite positive definite metric spaces when restricting to spaces that are clustered in a canonical neighborhood type. It leverages a geometric interpretation of magnitude via similarity embeddings and circumradius in Euclidean space, enabling precise control of magnitude through the circumradius $\rho_Y$ of the embedding. A sharp, if-and-only-if main result ties the convergence of magnitudes to the clustering type via $\|r\|_1$ and the cardinality $k$ of the limit space, with $|X_n|\to|X|$ exactly when $\|r\|_1 \le 2$ (any $k$) or $\|r\|_1=3$ and $k=1$. The paper also provides explicit counterexamples to show that relaxing these conditions can destroy convergence, thereby clarifying when magnitude behaves continuously under perturbations and offering a constructive approach to generate further counterexamples. These findings connect magnitude to Euclidean geometry through similarity embeddings, enriching both the theoretical understanding and potential data-analytic applications of magnitude in finite metric spaces.
Abstract
The magnitude of metric spaces does not to appear to possess a simple, convenient continuity property, and previous studies have presented affirmative results under additional constraints or weaker notions, as well as counterexamples. In this vein, we discuss the continuity of magnitude of finite positive definite metric spaces with respect to the Gromov-Hausdorff distance, but with a restriction of the domain based on a canonical partition of a sufficiently small neighborhood of a finite metric space. The main theorem of this article specifies the part of the partition for which, for a convergent sequence of finite metric spaces lying in the part, the magnitude converges to that of the limit. This study takes advantage of the geometric interpretation of magnitude as the circumradius of the corresponding finite Euclidean subset, called similarity embedding, as recently proposed by other studies. Such a transformation is especially useful for constructing counterexamples as we can depend on Euclidean geometric intuition.