Decomposition of pointwise finite-dimensional persistence modules

2012/10/02 by William Crawley-Boevey, Crawley-Boevey, William · 32 citations
Mathematics · #16G20 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:16G20

paper · pdf · doi:10.48550/arxiv.1210.0819

Changes in v3 (at request of a referee): more discussion of persistent homology and barcodes; some results mentioned in the introduction now formulated as corollaries

arxiv created 2014/07/29 · arxiv updated 2014/07/30

Abstract

We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on images and kernels.

Cited by

• Topology of Multi-species Localization
• Geometric Obstructions in Finsler Spaces and Torsion-Free Persistent Homology
• Persistent Cycle Representatives and Generalized Landscapes for Codimension 1 Persistent Homology
• Geometric realizations of Ringel-Hall algebras of continuous quivers of type A
• Continuous persistence landscapes
• The persistent homology of the Linial-Meshulam process
• Persistent Cost of Lipschitz Maps
• Interval Decomposition of Infinite Persistence Modules over a Principal Ideal Domain
• Fast Queries of Fibered Barcodes
• Precision-Graded Cohomology and Arithmetic Persistence for Network Sheaves
• Stability of persistent path homology of path complexes
• A Schauder Basis for Multiparameter Persistence
• Decomposing multipersistence modules using functor calculus
• Interleaving Distance as an Edit distance
• A rank-based distance for interval modules and Wasserstein stability of persistence landscapes
• Persistence modules induced by inner functions
• Universal topological statistics on triangulated singular spaces
• Interval Decompositions for Multipersistence Modules over Finite Posets and Robustness of Sheaf Data on Simplicial Complexes
• Limit Theorems for Verbose Persistence Diagrams
• Intrinsic Bottleneck Distance for Merge Trees
• Negative Calabi-Yau discrete cluster categories via Nakayama representations and persistence theory
• Kan Approximations of the Persistent Homology Transform
• Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack
• Interval-sphere model structures
• Equivalence of Landscape and Erosion Distances for Persistence Diagrams
• An isometry theorem for persistent homology of circle-valued functions
• Persistent homology of function spaces
• Coarse nodal counts on sub-Riemannian manifolds
• From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks
• Block-Decomposition for 3-Parameter Persistence Modules
• Barcode entropy for Reeb flows on contact manifolds with Liouville fillings
• Braiding vineyards

Related

• Decomposition of persistence modules
• On interval decomposition of persistence modules
• Interval Decomposition of Persistence Modules over a Principal Ideal Domain
• Interval Decomposition of Infinite Persistence Modules over a Principal Ideal Domain
• Interval Decomposition of Infinite Zigzag Persistence Modules