Stability for layer points
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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 v⃗-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 π_0 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 π_0 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 π_0 L_-,k(Y) to the layer points for π_0 L_-,0(X).
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