Persistence modules: Algebra and algorithms

Abstract

Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important part in the analysis and computation of persistent homology. In this paper we present a number of facts about persistence modules; ranging from the well-known but under-utilized to the reconstruction of techniques to work in a purely algebraic approach to persistent homology. In particular, the results we present give concrete algorithms to compute the persistent homology of a simplicial complex with torsion in the chain complex.

28 pages, submitted to Mathematics of Computation

References in corpus (2)

• Dualities in persistent (co)homology
• A spectral sequence for parallelized persistence

Cited by in corpus (5)

• Induced Matchings and the Algebraic Stability of Persistence Barcodes
• Randomly Weighted $d-$complexes: Minimal Spanning Acycles and Persistence Diagrams
• Sketches of a platypus: persistent homology and its algebraic foundations
• U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology
• Aspects of an internal logic for persistence