2021/09/03 by Katharine Adamyk, Adamyk, Katharine L. M.
Computer Science · Mathematics · Medicine · #55N31 #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Statistics Theory (math.ST) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2109.01701
openalex publication_date 2021/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the first half this paper, we generalize the theory of layer points for Lesnick- (or degree-Rips-) complexes to the more general context of v-hierarchical clusterings. Layer points provide a compressed description of a hierarchical clustering by recording only the points where a cluster changes. For multi-parameter hierarchical clusterings we consider both a global notion of layer points and layer points in the direction of a single parameter. An interleaving of hierarchical clusterings of the same set induces an interleaving of global layer points. In the particular, we consider cases where a hierarchical clustering of a finite metric space, Y, is interleaved with a hierarchical clustering of some sample X ⊆ Y. In the second half, we focus on the hierarchical clustering π0 L-,k(Y) for some finite metric space Y. When X ⊆ Y satisfies certain conditions guaranteeing X is well dispersed in Y and the points of Y are dense around X, there is an interleaving of layer points for π0 L-,k(Y) and a truncated version of L-,0(X) = V-(X). Under stronger conditions, this interleaving defines a retract from the layer points for π0 L-,k(Y) to the layer points for π0 L-,0(X).