Degenerating families of dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are $p$-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.

💡 Research Summary

The paper presents a novel mathematical framework for dendrograms, the hierarchical trees widely used in data analysis, by interpreting them as ultrametric spaces and embedding them into the non‑Archimedean geometry of p‑adic numbers. The authors begin by recalling that an ultrametric distance satisfies the strong triangle inequality d(x,z) ≤ max{d(x,y), d(y,z)}. This property is precisely the one exhibited by the p‑adic absolute value on the field ℚₚ, which immediately suggests a canonical p‑adic representation of any finite dendrogram. By adjoining a point at infinity to the set of leaves, the dendrogram becomes a complete rooted tree. The authors prove that this completed tree is isomorphic to a finite subtree of the Bruhat‑Tits tree associated with the projective line ℙ¹(ℚₚ). In the Bruhat‑Tits tree each vertex corresponds to a homothety class of ℤₚ‑lattices in ℚₚ², and edges encode inclusion relations; the ultrametric distance between two leaves is exactly the graph distance in this tree, scaled by logₚ.

Having established this geometric identification, the paper explores two major consequences. First, it connects families of dendrograms to classical moduli spaces in algebraic geometry. The space M₀,ₙ of stable n‑pointed rational curves can be realized over ℚₚ, and a dendrogram with n leaves determines a point of M₀,ₙ(ℚₚ) by placing the leaves as marked points on ℙ¹(ℚₚ). Varying the dendrogram continuously corresponds to moving along a p‑adic analytic path in M₀,ₙ(ℚₚ). Degenerations of the dendrogram—where several leaves coalesce at the same ultrametric level—are precisely the boundary points of the moduli space, where the underlying curve acquires nodal singularities. Thus the parameter space for dendrogram families is naturally a p‑adic analytic space, and the rich toolkit of deformation theory and rigid geometry becomes available for studying hierarchical clustering dynamics.

Second, the authors develop a stochastic classification scheme that respects the non‑Archimedean metric. Traditional Bayesian clustering assumes Euclidean distances and Gaussian likelihoods; here the likelihood of a data point belonging to a cluster is defined via a p‑adic probability measure that decays exponentially with the ultrametric distance to the cluster’s centroid. Because p‑adic distances are hierarchical by construction, the resulting posterior probabilities inherit a tree‑structured sparsity: points far from a cluster in the ultrametric sense receive negligible mass, while points within the same ultrametric ball share similar probabilities. The paper demonstrates on synthetic high‑dimensional data that this p‑adic Bayesian model captures multi‑scale cluster structures more faithfully than Euclidean counterparts, especially when clusters are nested or overlapping in a non‑linear fashion.

The final technical contribution concerns the “hidden part” of a dendrogram, defined as the subcomplex consisting of internal nodes that do not correspond to observed data points. By viewing the completed dendrogram as a simplicial complex, the hidden part becomes a compact 1‑dimensional cell complex. The authors compute its Betti numbers β₀ and β₁ using combinatorial Morse theory on the Bruhat‑Tits tree. They show that β₁, the number of independent cycles, grows with the degree of dendrogram degeneration: when many leaves share the same ultrametric level, the hidden subcomplex acquires additional loops. This topological invariant provides a quantitative measure of the intrinsic complexity and uncertainty of the hierarchical structure, offering a new descriptor for applications such as phylogenetics, document clustering, or any domain where hierarchical models are employed.

In summary, the paper bridges hierarchical clustering, p‑adic number theory, algebraic geometry, and topological data analysis. By embedding dendrograms into the Bruhat‑Tits tree, it furnishes a p‑adic analytic parameter space for families of trees, introduces a non‑Archimedean Bayesian classification framework, and supplies a topological invariant for the hidden structure of dendrograms. These insights open avenues for rigorous deformation studies of clustering algorithms, for exploiting p‑adic analytic continuation in model selection, and for integrating ultrametric topology into modern data‑science pipelines.