TY - GEN

T1 - Induced matchings of barcodes and the algebraic stability of persistence

AU - Bauer, Ulrich

AU - Lesnick, Michael

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2014

Y1 - 2014

N2 - We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).

AB - We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).

KW - Induced matching

KW - Interleaving distance

KW - Persistence module

UR - http://www.scopus.com/inward/record.url?scp=84904427522&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904427522&partnerID=8YFLogxK

U2 - 10.1145/2582112.2582168

DO - 10.1145/2582112.2582168

M3 - Conference contribution

AN - SCOPUS:84904427522

SN - 9781450325943

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 355

EP - 364

BT - Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014

PB - Association for Computing Machinery

T2 - 30th Annual Symposium on Computational Geometry, SoCG 2014

Y2 - 8 June 2014 through 11 June 2014

ER -