On the Stability of Interval Decomposable Persistence Modules

The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant $$2n-1$$ 2 n - 1 that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for $$n=2$$ n = 2 . We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.

The Reeb Graph Edit Distance is Universal

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.

Approximation of metric spaces by Reeb graphs: Cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds

For a connected locally path-connected topological space X and a continuous function f on it such that its Reeb graph Rf is a finite topological graph, we show that the cycle rank of Rf, i.e., the first Betti number b1(Rf), in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental group ?1(X), the condition of local path-connectedness being important since generally b1(Rf) can even exceed b1(X). We give some practical methods for calculating the co-rank of ?1(X) and a closely related value, the isotropy index. We apply our bound to improve upper bounds on the distortion of the Reeb quotient map, and thus on the Gromov-Hausdorff approximation of the space by Reeb graphs, for the distance function on a compact geodesic space and for a simple Morse function on a closed Riemannian manifold. This distortion is bounded from below by what we call the Reeb width b(M) of a metric space M, which guarantees that any real-valued continuous function on M has large enough contour (connected component of a level set). We show that for a Riemannian manifold, b(M) is non-zero and give a lower bound on it in terms of characteristics of the manifold. In particular, we show that any real-valued continuous function on a closed Euclidean unit ball E of dimension at least two has a contour C with diam(C??E)??3.