Generalizations of the Rips filtration for quasi-metric spaces with persistent homology stability results
Generalizations of the Rips filtration for quasi-metric spaces with persistent homology stability results
复制标题
具有持久同源稳定性结果的准度量空间 Rips 过滤的推广
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发表时间：
2016
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通讯作者：
Katharine Turner
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作者：
Katharine Turner
有限度量空间上的RIPS滤子及其相应的持久同调是拓扑数据分析中总结数据“形状”的重要方法。对它们的使用至关重要的是稳定性结果，即如果$X$和$Y$是有限度量空间，则由Rips过滤构造的持久性图、条形码或持久性模块之间的(瓶颈)距离由$2d_{GH}(X，Y)$(其中$d_{GH}$是Gromov-Hausdorff距离)限定。利用距离函数的不对称性，我们构造了四种不同的结构，类似于Rips滤波，它们捕捉了关于拟度量空间的不同信息。第一种方法是单参数对象族，其中对于拟度量空间$X$和[0，1]$中的$a，我们有单形复形$\\mathcal{R}^a(X)_t\}{t\in[0，\inty)}$的滤子，其中$\mathcal{R}^a(X)_t$是包含边$[x，y]$的团复形，无论何时$a\min\{d(x，y)，d(y，x)\}+(1-a)\max\{d(x，y)，D(y，x)\}\leq t$。第二种方法是构造一个有序元组复合体的滤子，其中元组$(x_0，x_2，x_p)\in数学{R}^{dir}(X)_t$if$d(x_i，x_j)\leq t$.当应用于度量空间时，我们的前两种方法都与正常的Rips滤波相一致。第三和第四种方法使用有向图的相关过滤，其中当$d(x，y)\leq t$时，$x\to y$包含在$D(X)_t$中。我们的第三种方法使用图$D(X)_t$的连通组件构建持久性模块。我们的第四种方法使用有向图$D_t$来使用偏序集拓扑创建偏序集(其中，如果存在从$x$到$y$的路径，则为$x\leq y$)和相应的持久性模块的过滤。
Rips filtrations over a finite metric space and their corresponding persistent homology are prominent methods in Topological Data Analysis to summarize the "shape" of data. Crucial to their use is the stability result that says if $X$ and $Y$ are finite metric space then the (bottleneck) distance between persistence diagrams, barcodes or persistence modules constructed by the Rips filtration is bounded by $2d_{GH}(X,Y)$ (where $d_{GH}$ is the Gromov-Hausdorff distance). Using the asymmetry of the distance function we construct four different constructions analogous to the Rips filtration that capture different information about the quasi-metric spaces. The first method is a one-parameter family of objects where, for a quasi-metric space $X$ and $a\in [0,1]$, we have a filtration of simplicial complexes $\{\mathcal{R}^a(X)_t\}_{t\in [0,\infty)}$ where $\mathcal{R}^a(X)_t$ is clique complex containing the edge $[x,y]$ whenever $a\min \{d(x,y), d(y,x) \}+ (1-a)\max \{d(x,y), d(y,x)\}\leq t$. The second method is to construct a filtration $\{\mathcal{R}^{dir}(X)_t\}$ of ordered tuple complexes where tuple $(x_0, x_2, \ldots x_p)\in \mathcal{R}^{dir}(X)_t$ if $d(x_i, x_j)\leq t$ for all $i\leq j$. Both our first two methods agree with the normal Rips filtration when applied to a metric space. The third and fourth methods use the associated filtration of directed graphs $\{D(X)_t\}$ where $x\to y$ is included in $D(X)_t$ when $d(x,y)\leq t$. Our third method builds persistence modules using the the connected components of the graphs $D(X)_t$. Our fourth method uses the directed graphs $D_t$ to create a filtration of posets (where $x\leq y$ if there is a path from $x$ to $y$) and corresponding persistence modules using poset topology.