Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

The reflection distance between zigzag persistence modules

The reflection distance between zigzag persistence modules By invoking the reflection functors introduced by Bernstein et al. (Russ Math Surv 28(2):17–32, 1973), in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the $$\ell ^1$$ ℓ 1 -bottleneck distance between their respective persistence diagrams. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

The reflection distance between zigzag persistence modules

Download PDF
35 pages

Loading next page...
 
/lp/springer-journals/the-reflection-distance-between-zigzag-persistence-modules-8znYbgzEqt
Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Algebraic Topology; Computational Science and Engineering; Mathematical and Computational Biology
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-019-00031-0
Publisher site
See Article on Publisher Site

Abstract

By invoking the reflection functors introduced by Bernstein et al. (Russ Math Surv 28(2):17–32, 1973), in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the $$\ell ^1$$ ℓ 1 -bottleneck distance between their respective persistence diagrams.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: Jul 27, 2019

References

  • Infinite matching theory
    Aharoni, R
  • Robust spatial memory maps encoded by networks with transient connections
    Babichev, A; Morozov, D; Dabaghian, Y
  • Coxeter functors and Gabriel’s theorem
    Bernstein, IN; Gel’fand, IM; Ponomarev, VA
  • Handbook of Categorical Algebra: Volume 1, Basic Category Theory. Cambridge Textbooks in Linguis
    Borceux, F; Rota, GC; Doran, B; Flajolet, P; Lam, TY; Lutwak, E; Ismail, M
  • Algebraic stability of zigzag persistence modules
    Botnan, M; Lesnick, M
  • Categorification of persistent homology
    Bubenik, P; Scott, JA
  • Metrics for generalized persistence modules
    Bubenik, P; Silva, V; Scott, J
  • Zigzag persistence
    Carlsson, G; Silva, V
  • The Structure and Stability of Persistence Modules
    Chazal, F; Silva, V; Glisse, M; Oudot, S
  • The importance of forgetting: limiting memory improves recovery of topological characteristics from neural data
    Chowdhury, S; Dai, B; Mémoli, F
  • Stability of persistence diagrams
    Cohen-Steiner, D; Edelsbrunner, H; Harer, J
  • Lipschitz functions have $$L_{p}$$ L p -stable persistence
    Cohen-Steiner, D; Edelsbrunner, H; Harer, J; Mileyko, Y
  • An incremental algorithm for betti numbers of simplicial complexes on the 3-sphere
    Delfinado, CJA; Edelsbrunner, H
  • Computational Topology: An Introduction
    Edelsbrunner, H; Harer, J
  • A distance for similarity classes of submanifolds of a euclidean space
    Frosini, P
  • Unzerlegbare darstellungen I
    Gabriel, P
  • The theory of the interleaving distance on multidimensional persistence modules
    Lesnick, M
  • Categories for the Working Mathematician
    Mac Lane, S
  • Theory of Graphs
    Ore, O
  • Persistence Theory: From Quiver Representations to Data Analysis
    Oudot, SY
  • Category Theory in Context
    Riehl, E
  • Towards computing homology from finite approximations
    Robins, V
  • Cantor, Schröder, and Bernstein in orbit
    Schweizer, B
  • Computing persistent homology
    Zomorodian, A; Carlsson, G