Stability of Persistence Diagrams

@article{CohenSteiner2007StabilityOP,
  title={Stability of Persistence Diagrams},
  author={David Cohen-Steiner and Herbert Edelsbrunner and John Harer},
  journal={Discrete \& Computational Geometry},
  year={2007},
  volume={37},
  pages={103-120}
}
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes. 
View on ACM
link.springer.com
911 Citations
Highly Influential Citations
92
Background Citations
510
Methods Citations
150
Results Citations
23

Figures from this paper

911 Citations

Persistent homology for functionals
. We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in…
Stability of persistence spaces of vector-valued continuous functions
TLDR
The persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram and its main result is its stability under function perturbations: any change invector-valued functions implies a not greater change in the Hausdorff distance between their persistence spaces.
Universality of the Bottleneck Distance for Extended Persistence Diagrams
TLDR
The question whether the bottleneck distance is the largest possible stable distance, providing an affirmative answer, is addressed.
Stability of persistent homology for hypergraphs
In topological data analysis, the stability of persistent diagrams gives the foundation for the persistent homology method. In this paper, we use the embedded homology and the homology of associated…
Stability of persistent homology for hypergraphs
In topological data analysis, the stability of persistent diagrams gives the foundation for the persistent homology method. In this paper, we use the embedded homology and the homology of associated…
Hausdorff Stability of Persistence Spaces
TLDR
The persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense and the main result is its stability under function perturbations.
Multiparameter Persistence Diagrams
TLDR
This paper studies the birth and death of cycles in a multifiltration of a chain complex with the goal of producing a persistence diagram that satisfies bottleneck stability.
Limit theorems for persistence diagrams
The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of…
A Continuous Theory of Persistence for Mappings Between Manifolds
Using sheaf theory, I introduce a continuous theory of persistence for mappings between compact manifolds. In the case both manifolds are orientable, the theory holds for integer coefficients. The…
The Persistent Homology of a Self-Map
TLDR
The persistence of the eigenspaces is defined, effectively introducing a hierarchical organization of the map in a continuous self-map and the induced endomorphism on homology.
...
...

References

SHOWING 1-10 OF 66 REFERENCES
Stratified Morse theory
Suppose that X is a topological space, f is a real valued function on X, and c is a real number. Then we will denote by X ≤c the subspace of points x in X such that f(x)≤c. The fundamental problem of…
Topological Persistence and Simplification
TLDR
Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
Elements of algebraic topology
Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in…
Inequalities for the curvature of curves and surfaces
TLDR
The difference between the total mean curvatures of two closed surfaces in R3 is bound in terms of their total absolute curvatures and the Fréchet distance between the volumes they enclose using a combination of methods from algebraic topology and integral geometry.
Computing persistent homology
TLDR
The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.
Persistence barcodes for shapes
TLDR
This paper initiates a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects, obtaining a shape descriptor, called a barcode, that is a finite union of intervals.
Morse theory
Topology on L a, b : Fix a broken λ λ , ... , λ . Its neighborhood consists of its deformations and smoothings. Key: every smooth trajectory has R symmetry. Can reduce to a level set in a Morse chart…
Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples
TLDR
It is proved that under quite general assumptions one can deduce the topology of a bounded open set in Rn from a Hausdorff distance approximation of it and the weak feature size (wfs) is introduced that generalizes the notion of local feature size.
Topological estimation using witness complexes
TLDR
This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object, and produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology.
Surface reconstruction by Voronoi filtering
TLDR
A simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points that uses Voronoi vertices to remove triangles from the Delaunay triangulation is given.
...
...