The Rips filtration over a finite metric space and its corresponding
persistent homology are prominent methods in TDA to summarise the shape of
data. Crucial to their use is the bottleneck stability result. A generalisation
of the Rips filtration to any symmetric function $f:X\times X\to \mathbb{R}$
was defined by Chazal, De Silva and Oudot, and they showed it was stable with
respect to the correspondence distortion distance. Allowing asymmetry, we
consider four different persistence modules. The first method is through
symmetrisation. For $a\in [0,1]$ we can construct a symmetric function
$sym_a(f)(x,y)=a \min \{d(x,y), d(y,x)\}+ (1-a)\max \{d(x,y), d(y,x)\}$. We can
then follow the apply the standard theory for symmetric functions and get
stability as a corollary. The second method is to construct a filtration of
ordered tuple complexes where tuple $(x_0, x_2, \ldots x_p)\in Rips^{dir}(X)_t$
if $d(x_i, x_j)\leq t$ for all $i\leq j$. These two methods have the same
persistent homology as the standard Rips filtration when applied to a metric
space, or more generally to a symmetric function. We consider two constructions
using a filtration of directed graphs. We have directed graphs $\{D(X)_t\}$,
where directed edges $x\to y$ are included in $D(X)_t$ whenever
$\max\{f(x,y),f(x,x),f(y,y)\}\leq t$. From this we construct a preorder where
$x\leq y$ if there is a path from $x$ to $y$ in $D(X)_t$. We build persistence
modules using the strongly connected components of the graphs $D(X)_t$, which
are the equivalence classes of the associated preorders. We consider
persistence modules using a generalisation of poset topology to preorders. The
Gromov-Hausdorff distance can be extended as a correspondence distortion
distance to set-function pairs. We prove that all these new constructions enjoy
the same stability results.

研究使用对称函数构建的 Rips 过滤器在特征提取与拓扑数据分析领域的表现，证明其在对称函数上的稳定性，并探讨了对称化方法、有序元组复合物方法、以及基于有向图的方法，进一步构建了包括预序拓扑学在内的新的持久化模块，都具有同样的稳定性