Proximity of persistence modules and their diagrams

@inproceedings{Chazal2009ProximityOP,
  title={Proximity of persistence modules and their diagrams},
  author={Fr{\'e}d{\'e}ric Chazal and David Cohen-Steiner and Marc Glisse and Leonidas J. Guibas and Steve Oudot},
  booktitle={SCG '09},
  year={2009}
}
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… 

Figures from this paper

Higher Interpolation and Extension for Persistence Modules
TLDR
A coherence criterion is isolated which guarantees the extensibility of non-expansive maps into this space of persistence modules across embeddings of the domain to larger ambient metric spaces, allowing Kan extensions to provide the desired extensions.
Metric Geometry of Spaces of Persistence Diagrams
. Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence
Geometry in the space of persistence modules
TLDR
It is shown that the relationship between the Cech and Rips complexes is governed by certain `coherence' conditions on the corresponding families of interleavings or matchings in the spaces of persistence modules and diagrams.
Rank-based persistence
TLDR
This work gives axioms for a generalized rank function on objects in a target category, so that functors to that category induce persistence functions and proves the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian categories.
Approximations of 1-dimensional intrinsic persistence of geodesic spaces and their stability
  • Žiga Virk
  • Mathematics
    Revista Matemática Complutense
  • 2018
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply
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.
Correspondence Modules and Persistence Sheaves: A Unifying Framework for One-Parameter Persistent Homology
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are
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.
Algebraic Stability of Zigzag Persistence Modules
TLDR
This paper functorially extends each zigzag persistence module to a two-dimensional persistence module, and establishes an algebraic stability theorem for these extensions, which yields a stability result for free two- dimensional persistence modules.
The Persistent Homotopy Type Distance
TLDR
This work proves that dHT extends the L-infty distance and dNP in two ways and shows that, appropriately restricting the category of objects to which dHT applies, it can be made to coincide with the other two distances.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 31 REFERENCES
The Stability of Persistence Diagrams Revisited
TLDR
New stability results for the persistence diagrams that lead to new applications in topological and geometric data analysis are proved.
Homological illusions of persistence and stability
In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of
Extending Persistence Using Poincaré and Lefschetz Duality
TLDR
An algebraic formulation is given that extends persistence to essential homology for any filtered space, an algorithm is presented to calculate it, and how it aids the ability to recognize shape features for codimension 1 submanifolds of Euclidean space is described.
The theory of multidimensional persistence
TLDR
This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.
Persistence-sensitive simplification of functions on 2-manlfolds
We continue the study of topological persistence [5] by investigating the problem of simplifying a function f in a way that removes topological noise as determined by its persistence diagram [2]. To
Persistence-sensitive simplication of functions on surfaces in linear time
Persistence provides a way of grading the importance of homological features in the sublevel sets of a real-valued function. Following the definition given by Edelsbrunner, Morozov and Pascucci, an
Stability of persistence diagrams
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
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.
Analysis of scalar fields over point cloud data
TLDR
This work introduces a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated and derives a series of algorithms for the analysis of scalar fields from point cloud data.
Persistence-sensitive simplification functions on 2-manifolds
TLDR
It is proved that for functions <i>f</i> on a 2-manifold such ε-simplification exists, and an algorithm to construct them in the piecewise linear case is given.
...
1
2
3
4
...