Rips filtrations for quasimetric spaces and asymmetric functions with stability results.

*(English)*Zbl 1434.55002The stability of persistent homology is well-established for metric spaces. Roughly speaking, it says that if two metric spaces are close to each other, then their persistent homology barcodes are also close to each other. This is a desirable property to have in data analysis contexts, where small perturbations of measurements are to be expected, and should not drastically alter one’s analysis. This paper studies the stability of persistent homology in the context of non-symmetric metric spaces, where the directed distance from point \(x\) to \(x'\) need not equal the directed distance from \(x'\) to \(x\).

In more detail, let \(X\) be a set equipped with a function \(d\colon X\times X\to \mathbb{R}\). We say that \(d\) is symmetric if \(d(x,x')=d(x',x)\) for all \(x,x'\in X\). One can think of a symmetric function \(d\) as a metric, even though it need not satisfy the triangle inequality. For \(d\) symmetric, the Vietoris-Rips complex of \(d\) at scale \(r\) has \(X\) as its vertex set, and a finite subset \(\sigma \subseteq X\) as a simplex if \(d(x,x')\le r\) for all \(x,x'\in \sigma\). If \(d\) and \(\tilde{d}\) are two symmetric functions, then the stability theorem in persistent homology states that the interleaving distance between the persistent homology modules of the Vietoris-Rips complexes for \(d\) and \(\tilde{d}\) is bounded from above by twice their Gromov-Hausdorff distance.

This paper considers four different adaptations of the above paragraph in the asymmetric setting, where \(d\colon X\times X\to \mathbb{R}\) is no longer assumed to be symmetric. The first two adaptations, roughly speaking, proceed by producing a symmetric function from an asymmetric one, and then applying the standard stability theorem. The third adaptation instead constructs an ordered tuple complex, or a directed complex, from the asymmetric function \(d\). The authors produce an analogous stability theorem bounding the persistent homology modules of the directed Vietoris-Rips complexes by a directed analogue of the Gromov-Hausdorff distance. The fourth adaptation functions in the setting of a generalization of poset topologies to preorders. These last two adaptations cannot be recovered via symmetrized techniques.

In more detail, let \(X\) be a set equipped with a function \(d\colon X\times X\to \mathbb{R}\). We say that \(d\) is symmetric if \(d(x,x')=d(x',x)\) for all \(x,x'\in X\). One can think of a symmetric function \(d\) as a metric, even though it need not satisfy the triangle inequality. For \(d\) symmetric, the Vietoris-Rips complex of \(d\) at scale \(r\) has \(X\) as its vertex set, and a finite subset \(\sigma \subseteq X\) as a simplex if \(d(x,x')\le r\) for all \(x,x'\in \sigma\). If \(d\) and \(\tilde{d}\) are two symmetric functions, then the stability theorem in persistent homology states that the interleaving distance between the persistent homology modules of the Vietoris-Rips complexes for \(d\) and \(\tilde{d}\) is bounded from above by twice their Gromov-Hausdorff distance.

This paper considers four different adaptations of the above paragraph in the asymmetric setting, where \(d\colon X\times X\to \mathbb{R}\) is no longer assumed to be symmetric. The first two adaptations, roughly speaking, proceed by producing a symmetric function from an asymmetric one, and then applying the standard stability theorem. The third adaptation instead constructs an ordered tuple complex, or a directed complex, from the asymmetric function \(d\). The authors produce an analogous stability theorem bounding the persistent homology modules of the directed Vietoris-Rips complexes by a directed analogue of the Gromov-Hausdorff distance. The fourth adaptation functions in the setting of a generalization of poset topologies to preorders. These last two adaptations cannot be recovered via symmetrized techniques.

Reviewer: Henry Adams (Fort Collins)

##### MSC:

55N31 | Persistent homology and applications, topological data analysis |

54E35 | Metric spaces, metrizability |

05C20 | Directed graphs (digraphs), tournaments |

06A11 | Algebraic aspects of posets |

55U10 | Simplicial sets and complexes in algebraic topology |

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\textit{K. Turner}, Algebr. Geom. Topol. 19, No. 3, 1135--1170 (2019; Zbl 1434.55002)

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