F. Chazal,
D. Cohen-Steiner, M. Glisse, L. J. Guibas, S. Y. Oudot. Proximity of
Persistence Modules and their Diagrams. Proc. 25th
ACM Sympos. on Comput. Geom., pages 237-246, 2009 (full version).
Topological persistence has proven to be a key concept for the study
of real-valued functions defined over topological spaces. Its validity
relies on the fundamental property that the persistence diagrams of
nearby functions are close. However, existing stability
results are restricted to the case of continuous functions defined over
triangulable spaces.
In this paper, we present new stability results that do not suffer
from the above restrictions. Furthermore, by working at an algebraic
level directly, we make it possible to compare the persistence
diagrams of functions defined over different spaces, thus enabling a
variety of new applications of the concept of persistence. Along the
way, we extend the definition of persistence diagram to a larger
setting, introduce the notions of discretization of a persistence
module and associated pixelization map, define a proximity measure
between persistence modules, and show how to interpolate between
persistence modules, thereby lending a more analytic character to this
otherwise algebraic setting. We believe these new theoretical concepts
and tools shed new light on the theory of persistence, in addition to
simplifying proofs and enabling new applications.
@inproceedings{ccggo-ppmd-09 , author = "F. Chazal and D. Cohen-Steiner and M. Glisse and L. J. Guibas and S. Y. Oudot" , title = "Proximity of Persistence Modules and their Diagrams" , booktitle = "Proc. 25th Annu. Symposium on Computational Geometry" , pages = "237--246" , year = 2009 } |