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 kerf and cokerf. As an
immediate corollary, we obtain a new proof of the algebraic stability of
persistence, 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, and
yields a novel "single-morphism" characterization of the interleaving relation
on persistence modules.Comment: Expanded journal version, to appear in Journal of Computational
Geometry. Includes a proof that no definition of induced matching can be
fully functorial (Proposition 5.10), and an extension of our single-morphism
characterization of the interleaving relation to multidimensional persistence
modules (Remark 6.7). Exposition is improved throughout. 11 Figures adde