Generalisations of the Rips Filtration for quasi-metric spaces and asymmetric functions with corresponding stability results

Seminar/Forum

Generalisations of the Rips Filtration for quasi-metric spaces and asymmetric functions with corresponding stability results

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. For a finite metric space \(X \) and distance \(r \), the traditional Rips complex with parameter (r) is the flag complex whose vertices are the points in (X) and whose edges are \({[x,y]: d(x,y)\leq r} \). From considering how the homology of these complexes evolves as we increase \(r\) we can create persistence modules (and their associated barcodes and persistence diagrams). Crucial to their use is the stability result that says if \(X\) and \(Y\) are finite metric space then the bottleneck distance between 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 we construct four different constructions analogous to the persistent homology of the Rips filtration and show they also are stable with respect to a natural generalisation of the Gromov-Hasdorff distance called the correspondence distortion distance. These different constructions involve ordered-tuple homology, symmetric functions of the distance function, strongly connected components and poset topology.

Presenter

  • Dr Katharine Turner
    Dr Katharine Turner, ANU