论文标题
有限度量空间的矩阵不变
Matrix Invariants of Finite Metric Spaces
论文作者
论文摘要
有限的度量空间的特征是根据距离函数和三角形不等式的阳性定义的多面体锥。它们的分类基于相关的多面体锥的分解，称为“公制风扇”。 $ n $ - 点公制空间的完整分类仅适用于$ n \ le 6 $。随着类的数量随元素的数量而迅速增加，希望基于有限度量空间的某些不变式具有更粗的等效类分解。如果$（x，d）$是一个有限的公制空间，带有元素$ p_i $，并且具有距离函数$ d_ {ij} $，则$ p_i $的gromov产品定义为$δ__{ijk} = 1/2（d_}+d_}+d_ {ij}+d_ {ik {ik {ik} -d_ -d_ {jk} $ {jk}）$。假设$ p_i $的Gromov产品集具有独特的最小元素$δ_{ijk} $，则边缘的关联$ p_jp_k $ to $ p_i $定义了“ Gromov产品结构”。有限度量空间的“无吊坠”减少是通过删除对应于最小gromov产品$δ__{ijk} $的边缘$ p_jp_k $获得的图表。在目前的工作中，我们为$ n $ - 点公制空间上的Gromov产品结构$ s $定义了矩阵表示，$ n \ times n $矩阵$ g_s $。我们证明，如果两个度量空间具有gromov产品结构，可以通过索引的排列互相映射，那么它们的矩阵通过相应的置换矩阵相似。 $ g_s $的矩阵不变性用于定义Gromov产品结构的子类，并给出了其应用于$ n = 5 $和$ n = 6 $ - 点空间。
Finite metric spaces are characterized by a polyhedral cone defined in terms of the positivity of the distance functions and the triangle inequalities. Their classification is based on the decomposition of an associated polyhedral cone, called the "metric fan". The complete classification of $n$-point metric spaces is available only for $n\le 6$. As the number of classes increases rapidly with the number of elements, it is desirable to have coarser equivalence class decompositions based on certain invariants of finite metric spaces. If $(X,d)$ is a finite metric space with elements $P_i$ and with distance functions $d_{ij}$, the Gromov product at $P_i$ is defined as $Δ_{ijk}=1/2(d_{ij}+d_{ik}-d_{jk})$. Assuming that the set of Gromov product at $P_i$ has a unique smallest element $Δ_{ijk}$, the association of the edge $P_jP_k$ to $P_i$ defines the "Gromov product structure". The "pendant-free" reduction of the finite metric space is the graph obtained by removing the edges $P_jP_k$ corresponding to the minimal Gromov products $Δ_{ijk}$ at $P_i$. In the present work, we define a matrix representation for a Gromov product structure $S$ on an $n$-point metric space, by $n\times n$ matrix $G_S$. We prove that if two metric spaces have Gromov product structures that can be mapped to each other by a permutation of the indices, then their matrices are similar via the corresponding permutation matrix. Matrix invariants of $G_S$ are used to define subclasses of Gromov product structures and their application to $n=5$ and $n=6$-point spaces are given.
