New Stability Theory for Subdominant Ultrametric Under Sparse Perturbations

A recent study published on arXiv (2608.04014) presents a stability theory for the subdominant (minmax) ultrametric, a standard tree-like representation of dissimilarity matrices, which aligns with single-linkage clustering. Traditional stability limits in ℓ∞ or Gromov–Hausdorff contexts are inadequate for sparse perturbations that impact only a small number of pairwise distances. The researchers propose an ℓ0-type stability framework, revealing that sparse modifications influence solely the minimum spanning tree (MST). A change in a pairwise ultrametric value occurs only if its tree path intersects an altered edge or a newly exposed cut from an edited off-tree edge. This results in a precise per-edit exposed-cut score and a tree-only global envelope, establishing Hamming–Lipschitz bounds on the potential changes in ultrametric entries. The paper also confirms sharpness results, indicating that reliance on tree geometry is essential. The findings are theoretical, featuring straightforward proofs, and have significant implications for clustering stability and data analysis.