Discrete & Computational Geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Inequalities for the curvature of curves and surfaces
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Persistence-sensitive simplification functions on 2-manifolds
Proceedings of the twenty-second annual symposium on Computational geometry
The theory of multidimensional persistence
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Inferring Local Homology from Sampled Stratified Spaces
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Multidimensional Size Functions for Shape Comparison
Journal of Mathematical Imaging and Vision
Analysis of scalar fields over point cloud data
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Extending Persistence Using Poincaré and Lefschetz Duality
Foundations of Computational Mathematics
Homological illusions of persistence and stability
Homological illusions of persistence and stability
Zigzag persistent homology and real-valued functions
Proceedings of the twenty-fifth annual symposium on Computational geometry
Gromov-Hausdorff stable signatures for shapes using persistence
SGP '09 Proceedings of the Symposium on Geometry Processing
Topological inference via meshing
Proceedings of the twenty-sixth annual symposium on Computational geometry
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
Persistence-based clustering in riemannian manifolds
Proceedings of the twenty-seventh annual symposium on Computational geometry
An output-sensitive algorithm for persistent homology
Proceedings of the twenty-seventh annual symposium on Computational geometry
Local homology transfer and stratification learning
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
Persistence modules, shape description, and completeness
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
An output-sensitive algorithm for persistent homology
Computational Geometry: Theory and Applications
Stable Comparison of Multidimensional Persistent Homology Groups with Torsion
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Geometry in the space of persistence modules
Proceedings of the twenty-ninth annual symposium on Computational geometry
Comparison of persistent homologies for vector functions: From continuous to discrete and back
Computers & Mathematics with Applications
Persistence-Based Clustering in Riemannian Manifolds
Journal of the ACM (JACM)
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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.