The stability theorem for persistent homology is a central result in topological data
analysis. While the original formulation of the result concerns the persistence barcodes of
–valued
functions, the result was later cast in a more general algebraic form, in the language
of
persistence modules and
interleavings. We establish an analogue of this algebraic
stability theorem for zigzag persistence modules. To do so, we functorially extend
each zigzag persistence module to a two-dimensional persistence module, and
establish an algebraic stability theorem for these extensions. One part of our
argument yields a stability result for free two-dimensional persistence modules. As an
application of our main theorem, we strengthen a result of Bauer et al on the
stability of the persistent homology of Reeb graphs. Our main result also yields an
alternative proof of the stability theorem for level set persistent homology of Carlsson
et al.
Keywords
topological data analysis, persistent homology,
interleavings
