层点稳定性
Stability for layer points
摘要：
在本文的前半部分，我们将Lesnick-（或度-Rips-）复形的层点理论推广到更一般的$\vec{v}$ -层次聚类。层点通过仅记录聚类发生变化的点来提供分层聚类的压缩描述。对于多参数层次聚类，我们认为这是一个全球性的概念层点和层点的方向上的一个单一的参数。同一集合的分层聚类的交织引起全局层点的交织。特别是，我们考虑的情况下，一个有限的度量空间，$Y$的层次聚类，交错与一些样本$X \subseteq Y$的层次聚类。在后半部分中，我们重点研究了有限度量空间$Y$上的层次聚类算法$π_0 L_{-,k}(Y)$。当$X \subseteq Y$满足保证$X$在$Y$中良好分散并且$Y$的点在$X$周围密集的某些条件时，存在用于$π_0 L_{-,k}(Y)$和$L_{-,0}(X) = V_{-}(X)$的截断版本的层点的交织。在更强的条件下，这种交织定义了从$π_0 L_{-,k}(Y)$的层点到$π_0 L_{-,0}(X)$的层点的回退。
英文摘要：
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 $π_0 L_{-,k}(Y)$ 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 $π_0 L_{-,k}(Y)$ 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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