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. 

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References

SHOWING 1-10 OF 70 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
The λ-medial axis
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.
Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions
TLDR
The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance, and it is proved that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.
...
...