Induced Matchings of Barcodes and the Algebraic Stability of Persistence

Tuesday, November 12, 2013 - 1:30pm - 2:30pm
Lind 305
Michael Lesnick (University of Minnesota, Twin Cities)
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 of Persistence, a fundamental result
in the theory of persistent homology due originally to Chazal et al.,
building on work of Cohen-Steiner et al. In contrast to previous proofs of
the Algebraic Stability of Persistence, 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.

This is joint work with Ulrich Bauer.