Robust Geodesic Learning via Hierarchical Connectivity

Geodesic rank is a fundamental signal for manifold learning, but it is difficult to recover when low-dimensional structure is embedded in a high-dimensional noisy ambient space. Irrelevant ambient coordinates can dominate nearest-neighbor relations, making graph-based methods unstable. We study robust geodesic learning as the problem of recovering both geodesic rank and topological structure under such noise. We propose AGD-SS, a dissimilarity measure built from random agglomerative hierarchical clustering forests. For each subsample, AGD-SS projects query points onto the sampled dendrogram and aggregates merge heights in the subtree rooted at their lowest common ancestor, using hierarchical connectivity rather than raw ambient neighborhoods. This construction connects geodesic learning with minimum spanning trees and 0-dimensional persistent homology, and supports a stability analysis of the estimator under perturbation. Experiments on synthetic manifolds, non-Euclidean settings, the Drosophila connectome, and single-cell datasets show that AGD-SS preserves local and global geodesic rank while recovering topological structure under ambient noise. These results suggest that hierarchical connectivity provides a simple inductive bias for robust representation learning that preserves both geometry and topology.