A crucial step in the analysis of persistent homology is transformation data into an appropriate topological object (which, our case, a simplicial complex). Software packages for computing typically construct Vietoris--Rips or other distance-based complexes on point clouds because they are relatively easy to compute. We investigate alternative methods constructing and effects making associated ...

Persistent homology has widespread applications in computer vision and image analysis. This paper first motivates the use of persistent homology as a suitable tool to solve the problem of extracting global topological information from a discrete sample of points. The remainder of this paper develops the mathematical theory behind persistent homology. Persistent homology will be developed as an ...

Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct an...

In this paper, we establish a correspondence between the incremental algorithm for computing AT-models [8,9] and the one for computing persistent homology [6,14,15]. We also present a decremental algorithm for computing AT-models that allows to extend the persistence computation to a wider setting. Finally, we show how to combine incremental and decremental techniques for persistent homology co...

We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop ...

This paper presents methods to compare high order networks using persistence homology. High order networks induce well-founded homological features and the difference between networks is measured by the difference between the homological features. This is a reasonable approximation to a valid metric in the space of high order networks modulo permutation isomorphisms. The approximations succeed ...

Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbation...

The algebraic stability theorem for pointwise finite dimensional (p.f.d.) R-persistence modules is a central result in the theory of stability for persistence modules. We present a stability theorem for n-dimensional rectangle decomposable p.f.d. persistence modules up to a constant (2n− 1) that is a generalization of the algebraic stability theorem. We give an example to show that the bound ca...

In algebraic topology it is well-known that, using the Mayer-Vietoris sequence, the homology of a space X can be studied splitting X into subspaces A and B and computing the homology of A, B, A∩B. A natural question is to which an extent persistent homology benefits of a similar property. In this paper we show that persistent homology has a Mayer-Vietoris sequence that in general is not exact b...

We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduc...