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 $\pi_0 L_{-,0}(X)$.
翻译:在本文的前半部分, 我们将Lesnick- (或度- 里普斯- ) 复合层的层点理论概括为 $\ vec{v} $- 等级组群的更一般的背景 。 层点只记录组群变化的点。 对于多参数的等级组群, 我们同时考虑一个单一参数方向的多参数点点和层点的全球概念。 同一组的等级组群相互交错, 引发一个全球层点的中间值。 特别是, 我们考虑的情况是, 一个限定的基点( Y$) 的等级组群, 一个基点是 $X\ subsesesesec Y. 在后半部分, 我们侧重于某个限定的基点 $\ pi_ 0 L- k (Y) 。 当一个基点的 $X\ subsetc $ n_ x_ r_ lixx 的基点, 一个基点是这个基值的基点, 一个基值为 $x_ 美元, 一个基值的基值。