Rips filtrations for quasi-metric spaces and asymmetric functions with stability results

Abstract

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.