Erosion distance for generalized persistence modules

Abstract

The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer was recently generalized by Patel to the case of constructible persistence modules with values in a symmetric monoidal category with images. Patel also introduced a distance for persistence diagrams, the erosion distance. Motivated by this work, we extend the erosion distance to a distance of rank invariants of generalized persistence modules by using the generalization of the interleaving distance of Bubenik, de Silva, and Scott as a guideline. This extension of the erosion distance also gives, as a special case, a distance for multidimensional persistent homology groups with torsion introduced by Frosini. We show that the erosion distance is stable with respect to the interleaving distance, and that it gives a lower bound for the natural pseudo-distance in the case of sublevel set persistent homology of continuous functions.

Metrics and Stabilization in One Parameter Persistence

“…Contours appeared independently in [5] under the name superlinear families where they were used to define interleaving distances between generalized persistence modules. Since then generalized persistence modules have gathered some interest, see [11] and [15]. In this section we presented a variety of ways of constructing contours greatly enlarging [5], in which only the standard contour is given as a concrete example of a superlinear family.…”

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

“…Hence, we have that rk(M )(a , b ) ≤ rk(M )(a, b). This means that rk(M ) is a functor between its domain and codomain when regarded We have stability of the rank invariant: [59,Theorem 22] in more general setting, we provide a brief version of the proof here.…”

“…Remark 6.3. In order to compare the rank invariants, the author of [59] makes use of a generalization of the erosion distance in [58], which is denoted by d E (see Section C). It can be deduced that for the LHS of inequality (14) coincides with d E (rk(M ), rk(N )).…”

“…Note that since U is a subposet of R op × R, we can regard d E is a particular case of d I,2 from Section 3.2. The erosion distance is further generalized in [59].…”

“…been studied in various contexts and with various names such as Betti curve, feature counting function, etc, [2,28,34,35,42,62]. The rank function rk(M X ) of M X has also been extensively considered [17,19,51,58,59]. We observe that both of these functions (1) can themselves be computed in polynomial time, (2) can be compared to each other via the interleaving distance d Z I for integer-valued functions (see Section 3.2) and (3) are stable to perturbations of (X , d X (·)) under d dyn (Theorems 4.…”