Stability for layer points

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 $\vec{v}$-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 $X \subseteq Y$. In the second half, we focus on the hierarchical clustering $\pi_0 L_{-,k}(Y)$ for some finite metric space $Y$. When $X \subseteq Y$ 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 $\pi_0 L_{-,k}(Y)$ and a truncated version of $L_{-,0}(X) = V_{-}(X)$. Under stronger conditions, this interleaving defines a retract from the layer points for $\pi_0 L_{-,k}(Y)$ to the layer points for $\pi_0 L_{-,0}(X)$.