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

Seminar/Forum

107
Peter Hall
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.