Betti numbers in multidimensional persistent homology are stable functions (Q2857523)

Define the relation \(\prec\) on \({\mathbb R}^{n}\) by \(u \prec v\) iff \(u_{i} < v_{i}\) for all \(i = 1,\ldots,n\). Given a function \(\phi : X \rightarrow {\mathbb R}^{n}\) for any \(u \prec v\) one has an induced mapping of Čech homology \(\pi_{k}^{(u,v) }: H_{k}(X(\phi \prec u) \rightarrow H_{k}(X(\phi \prec v))\). The image of this map is called the multidimensional \(k\)th persistent homology group of \((X,\phi)\) at \((u,v)\). The authors show that if \(X\) is a triangulable space then these Betti numbers are finite. One can associate to the maps \(\phi\) a diagram in \(\Delta^{+} = \{(u,v) \in {\mathbb R}^{n} \times {\mathbb R}^{n}: u \prec v\}\) that represents the persistent Betti numbers. If the map \(\phi\) is perturbed then the diagram shifts and a distance between the diagrams can be defined. The main result of the paper is that these multidimensional persistent Betti numbers are stable in the sense that small perturbations in the function \(\phi\) induce only small changes in the Betti numbers. This is significant for the applications of persistent homology to the study of large data sets.