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M. Botnan, J. Curry, E. Munch (2020)
A Relative Theory of InterleavingsArXiv, abs/2004.14286
E. Munch, Anastasios Stefanou (2018)
The ℓ∞-Cophenetic Metric for Phylogenetic Trees as an Interleaving DistanceArXiv, abs/1803.07609
Martina Scolamiero, W. Chachólski, A. Lundman, Ryan Ramanujam, Sebastian Öberg (2015)
Multidimensional Persistence and NoiseFoundations of Computational Mathematics, 17
Ulrich Bauer, M. Botnan, Steffen Oppermann, Johan Steen (2019)
Cotorsion torsion triples and the representation theory of filtered hierarchical clusteringAdvances in Mathematics
Peter Bubenik, Alex Elchesen (2020)
Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spacesJournal of Applied and Computational Topology, 6
Vincent Divol, Théo Lacombe (2019)
Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transportJournal of Applied and Computational Topology, 5
Peter Bubenik, Jonathan Scott (2012)
Categorification of Persistent HomologyDiscrete & Computational Geometry, 51
Ezra Miller (2019)
Modules over posets: commutative and homological algebraarXiv: Commutative Algebra
M. Lesnick, Matthew Wright (2015)
Interactive Visualization of 2-D Persistence ModulesArXiv, abs/1512.00180
(2020)
Botnan and William Crawley-Boevey
F. Chazal, D. Cohen-Steiner, M. Glisse, L. Guibas, S. Oudot (2009)
Proximity of persistence modules and their diagrams
V. Silva, E. Munch, A. Patel (2015)
Categorified Reeb GraphsDiscrete & Computational Geometry, 55
Ezra Miller (2020)
Essential graded algebra over polynomial rings with real exponentsarXiv: Commutative Algebra
U Bauer (2015)
162J. Comput. Geom., 6
Ulrich Bauer, M. Lesnick (2013)
Induced matchings and the algebraic stability of persistence barcodesJ. Comput. Geom., 6
Alex McCleary, A. Patel (2018)
Bottleneck stability for generalized persistence diagramsProceedings of the American Mathematical Society
A. Thomas (2019)
Invariants and Metrics for Multiparameter Persistent Homology
U Bauer (2020)
107171Adv. Math., 369
D Cohen-Steiner, H Edelsbrunner, J Harer, Y Mileyko (2010)
Lipschitz functions have Lp\documentclass[12pt]{minimal}Found. Comput. Math., 10
(2020)
arXiv:2004.14286 [math.CT
A. Patel (2016)
Generalized persistence diagramsJournal of Applied and Computational Topology, 1
D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Yuriy Mileyko (2010)
Lipschitz Functions Have Lp-Stable PersistenceFoundations of Computational Mathematics, 10
H Krause (2015)
Krull?Schmidt categories and projective coversExpos. Math., 33
Peter Bubenik, V. Silva, Jonathan Scott (2017)
Interleaving and Gromov-Hausdorff distancearXiv: Category Theory
P. Gabriel (1972)
Unzerlegbare Darstellungen Imanuscripta mathematica, 6
Barbara Giunti, John Nolan, N. Otter, Lukas Waas (2021)
Amplitudes on abelian categories
D. Morozov, Kenes Beketayev, G. Weber (2013)
Interleaving Distance between Merge Trees
Peter Bubenik, Alex Elchesen (2019)
Universality of persistence diagrams and the bottleneck and Wasserstein distancesArXiv, abs/1912.02563
M. Botnan, W. Crawley-Boevey (2018)
Decomposition of persistence modulesProceedings of the American Mathematical Society
H. Krause (2014)
Krull-Schmidt categories and projective coversarXiv: Representation Theory
H 0 (Y )) = 2. Since H 0 (X) and H 0 (Y ) have identical dimension vectors
H. Harrington, N. Otter, H. Schenck, U. Tillmann (2017)
Stratifying Multiparameter Persistent HomologySIAM J. Appl. Algebra Geom., 3
(1968)
Pure and Applied Mathematics, vol. XIX
Harrington , Nina Otter , Hal Schenck , and Ulrike Tillmann . block Stratifying Mul - tiparameter Persistent Homology
S. Harker, M. Kramár, R. Levanger, K. Mischaikow (2018)
A comparison framework for interleaved persistence modulesJournal of Applied and Computational Topology, 3
Vincent Divol, Théo Lacombe (2019)
Understanding the Topology and the Geometry of the Persistence Diagram Space via Optimal Partial TransportArXiv, abs/1901.03048
M. Lesnick (2011)
The Theory of the Interleaving Distance on Multidimensional Persistence ModulesFoundations of Computational Mathematics, 15
A. Blumberg, M. Lesnick (2017)
Universality of the Homotopy Interleaving DistanceArXiv, abs/1705.01690
D. Cohen-Steiner, H. Edelsbrunner, J. Harer (2005)
Stability of Persistence DiagramsDiscrete & Computational Geometry, 37
Peter Bubenik, T. Vergili (2018)
Topological spaces of persistence modules and their propertiesJournal of Applied and Computational Topology, 2
Alex Elchesen, F. Mémoli (2018)
The reflection distance between zigzag persistence modulesJournal of Applied and Computational Topology, 3
P Bubenik, A Elchesen (2022)
Universality of persistence diagrams and the bottleneck and Wasserstein distancesComput. Geom., 105–106
P. Skraba, Katharine Turner (2020)
Wasserstein Stability for Persistence DiagramsarXiv: Algebraic Topology
B. Stenström (1975)
Rings of Quotients
(1970)
Bodo Pareigis. block Categories and functors. block Translated from the German
Ezra Miller (2020)
Primary decomposition over partially ordered groupsarXiv: Commutative Algebra
V. Silva, E. Munch, Anastasios Stefanou (2017)
Theory of interleavings on categories with a flowarXiv: Category Theory
Peter Bubenik, V. Silva, Jonathan Scott (2013)
Metrics for Generalized Persistence ModulesFoundations of Computational Mathematics, 15
Peter Bubenik, Nikola Milićević (2019)
Homological Algebra for Persistence ModulesFoundations of Computational Mathematics, 21
N. Popescu (1973)
Abelian categories with applications to rings and modules
I. Bucur, A. Deleanu, P. Hilton (1968)
Introduction to the Theory of Categories and Functors
W. Crawley-Boevey (2012)
Decomposition of pointwise finite-dimensional persistence modulesarXiv: Representation Theory
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
W Crawley-Boevey (2015)
Decomposition of pointwise finite-dimensional persistence modulesJ. Algebra Appl., 14
We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space. We use this distance to define Wasserstein distances, which coincide with the previously defined Wasserstein distances for one-parameter persistence modules. For one-parameter persistence modules, we also describe maps to and from an interval module, and we give a matrix reduction for monomorphisms and epimorphisms.
Journal of Applied and Computational Topology – Springer Journals
Published: Jun 1, 2023
Keywords: Distances for abelian categories; Persistence modules; Wasserstein distance; 55N31; 18E10
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