Offset filtration

Let X be a finite set in a metric space (M,d), and for any x\in X let B(x,\varepsilon) = \{y\in X \mid d(x,y) \leq \varepsilon \} be the closed ball of radius \varepsilon centered at x. Then the union X^{(\varepsilon)}:=\bigcup_{x\in X} B(x,\varepsilon) is known as the offset of X with respect to the parameter \varepsilon (or simply the \varepsilon-offset of X). By considering the collection of offsets over all \varepsilon \in [0,\infty) we get a family of spaces \mathcal O(X) := \{ X^{(\varepsilon)} \mid \varepsilon \in [0,\infty)\} where X^{(\varepsilon)}\subseteq X^{(\varepsilon^\prime)} whenever \varepsilon \leq \varepsilon^\prime. So \mathcal O(X) is a family of nested topological spaces indexed over \varepsilon, which defines a filtration known as the offset filtration on X. Note that it is also possible to view the offset filtration as a functor \mathcal O(X) : [0, \infty) \to \mathbf{Top} from the poset category of non-negative real numbers to the category of topological spaces and continuous maps. There are some advantages to the categorical viewpoint, as explored by Bubenik and others. == Properties ==

A standard application of the nerve theorem shows that the union of balls has the same homotopy type as its nerve, since closed balls are convex and the intersection of convex sets is convex. The nerve of the union of balls is also known as the Čech complex, which is a subcomplex of the Vietoris-Rips complex. Therefore the offset filtration is weakly equivalent to the Čech filtration (defined as the nerve of each offset across all scale parameters), so their homology groups are isomorphic. Although the Vietoris-Rips filtration is not identical to the Čech filtration in general, it is an approximation in a sense. In particular, for a set X \subset \mathbb R^d we have a chain of inclusions \operatorname{Rips}_\varepsilon(X) \subset \operatorname{Cech}_{\varepsilon^\prime}(X) \subset \operatorname{Rips}_{\varepsilon^\prime}(X) between the Rips and Čech complexes on X whenever \varepsilon^\prime / \varepsilon \geq \sqrt{2d/d+1}. In general metric spaces, we have that \operatorname{Cech}_\varepsilon(X) \subset \operatorname{Rips}_{2\varepsilon}(X) \subset \operatorname{Cech}_{2\varepsilon}(X) for all \varepsilon >0, implying that the Rips and Cech filtrations are 2-interleaved with respect to the interleaving distance as introduced by Chazal et al. in 2009. It is a well-known result of Niyogi, Smale, and Weinberger that given a sufficiently dense random point cloud sample of a smooth submanifold in Euclidean space, the union of balls of a certain radius recovers the homology of the object via a deformation retraction of the Čech complex. The offset filtration is also known to be stable with respect to perturbations of the underlying data set. This follows from the fact that the offset filtration can be viewed as a sublevel-set filtration with respect to the distance function of the metric space. The stability of sublevel-set filtrations can be stated as follows: Given any two real-valued functions \gamma, \kappa on a topological space T such that for all i\geq 0, the i\text{th}-dimensional homology modules on the sublevel-set filtrations with respect to \gamma, \kappa are point-wise finite dimensional, we have d_B (\mathcal B_i (\gamma), \mathcal B_i (\kappa)) \leq d_\infty (\gamma, \kappa) where d_B(-) and d_\infty(-) denote the bottleneck and sup-norm distances, respectively, and \mathcal B_i (-) denotes the i\text{th}-dimensional persistent homology barcode. While first stated in 2005, this sublevel stability result also follows directly from an algebraic stability property sometimes known as the "Isometry Theorem," A multiparameter extension of the offset filtration defined by considering points covered by multiple balls is given by the multicover bifiltration, and has also been an object of interest in persistent homology and computational geometry. ==References==