On Hamming–Lipschitz Type Stability of the Subdominant (Minmax) Ultrametric: Theory and Simple Proofs

TMLR Paper7150 Authors

Abstract: We study the subdominant (minmax) ultrametric as an operator on pairwise data. Prior stability results show that this operator is non-expansive under uniform perturbations in the supremum norm and in the Gromov–Hausdorff sense, but they say nothing about how widely sparse, targeted edits can ripple through the hierarchy. We close this gap with a pair-count Lipschitz theory in Hamming space: we bound how many ultrametric entries can change, regardless of their magnitudes. The analysis is routed through the \emph{minimum spanning tree} (MST), which encodes the ultrametric as path bottlenecks. Our first theorem proves a locality principle; only pairs whose MST path crosses an edited or newly exposed cut can change, so the impact is confined to a union of fundamental cut rectangles. Building on this, we derive an instance dependent $$\ell_0$$ type Lipschitz bound whose constant is determined entirely by the MST’s exposed cuts. We then show optimality by constructing cases where a single off-tree edit forces a quadratic number of changes, so no smaller universal constant is possible for our proposed Lipschitz constant. Finally, under a mild minimal-overlap condition, the upper bound on the number of changed entries of the ultrametric is order-tight, yielding a two-sided characterization of propagation. Conceptually, this advances a magnitude-versus-extent picture for ultrametric stability: classical results control how much entries move under uniform perturbation; our theory controls how far changes spread under sparse edits. Additionally, as a proof of concept, we derive a risk score from our Lipschitz constant that identifies vulnerable edges in the graph. We use this score to drive two case studies: vulnerability maps of deep embeddings of CIFAR-10, ImageNet-10, and STL-10, where targeted edits to high-score edges cause far larger ultrametric and clustering changes than random edits with the same budget, and fragility maps in a superpixel-based single image segmentation that highlight load-bearing boundaries.

Submission Type: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=NJS7QqvwKI&referrer=%5BAuthor%20Console%5D(%2Fgroup%3Fid%3DTMLR%2FAuthors%23your-submissions)

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Assigned Action Editor: ~Akshay_Rangamani1

Submission Number: 7150