Determine practical localization-graph sample-size and scale parameters

Determine actual values of the auxiliary sample sizes N(j) and localization-graph parameters ε₁(j) and ε₂(j) that guarantee convergence of shortest-path dissimilarities to the Riemannian distance on an unknown support manifold.

Background

Theorem SPDconverge establishes only the existence of sequences of auxiliary sample sizes and localization parameters for which the shortest-path dissimilarities converge in ratio to the Riemannian distance. Its conditions depend on unknown geometric quantities of the support manifold, including the minimum radius of curvature and minimum branch separation. Consequently, a practical implementation of the Isomap-style procedure lacks a demonstrated method for selecting the sample size and graph scales in a way that guarantees the theoretical convergence result.

References

It is important to recognize that, while Theorem~\ref{thm:SPDconverge} establishes the {\em existence}\/ of convergent sequences, it does not construct an algorithm that is guaranteed to converge to the desired limit. The stated conditions involve parameters of the support manifold, which is unknown. It is therefore unclear how to determine actual values of $N(j)$, $\epsilon_1(j)$, and $\epsilon_2(j)$ that guarantee the desired convergence.

— Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theory  (2609.19357 - Trosset et al., 16 Sep 2026) in Section 6, Discussion