Name
Abstract | ||
|---|---|---|
In this paper a parameterized generalization of a good cover filtration is introduced called an {epsilon}-good cover, defined as a cover filtration in which the reduced homology groups of the image of the inclusions between the intersections of the cover filtration at two scales {epsilon} apart are trivial. Assuming that one has an {epsilon}-good cover filtration of a finite simplicial filtration, we prove a tight bound on the bottleneck distance between the persistence diagrams of the nerve filtration and the simplicial filtration that is linear with respect to {epsilon} and the homology dimension. This bound is the result of a computable chain map from the nerve filtration to the space filtrationu0027s chain complexes at a further scale. Quantitative guarantees for covers that are not good are useful for when one is working a non-convex metric space, or one has more simplicial covers that are not the result of triangulations of metric balls. The Persistent Nerve Lemma is also a corollary of our theorem as good covers are 0-good covers. Lastly, a technique is introduced that symmetrizes the asymmetric interleaving used to prove the bound by shifting the nerve filtrationu0027s persistence module, improving the interleaving constant by a factor of 2. |
Year | Venue | Field |
|---|---|---|
2018 | arXiv: Algebraic Topology | Combinatorics,Parameterized complexity,Algebra,Ball (bearing),Filtration,Metric space,Reduced homology,Corollary,Lemma (mathematics),Interleaving,Mathematics |
DocType | Volume | Citations |
Journal | abs/1807.07920 | 0 |
PageRank | References | Authors |
0.34 | 7 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Nicholas J. Cavanna | 1 | 4 | 1.86 |
| Donald R. Sheehy | 2 | 72 | 13.98 |