Finiteness of rank invariants of multidimensional persistent homology groups

  title={Finiteness of rank invariants of multidimensional persistent homology groups},
  author={Francesca Cagliari and Claudia Landi},
Parametrized homology via zigzag persistence
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological
The Persistence Space in Multidimensional Persistent Homology
The persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense and a method to visualize topological features of a shape via persistence spaces is presented.
Persistent Homology and the Upper Box Dimension
A fractal dimension for a metric space based on the persistent homology of subsets of that space is introduced and hypotheses under which this dimension is comparable to the upper box dimension are exhibited.
Stable Comparison of Multidimensional Persistent Homology Groups with Torsion
A pseudo-distance dT is introduced that represents a possible solution to the present lack of a stable method to compare persistent homology groups with torsion, and the main theorem proves the stability of the new pseudo- distance with respect to the change of the filtering function.
The Persistent Homology of Random Geometric Complexes on Fractals
We study the asymptotic behavior of the persistent homology of i.i.d. samples from a d-Ahlfors regular measure on a metric space — one that satisfies uniform bounds of the form 1 c r ≤ μ (Br (x)) ≤ c
Geometry in the space of persistence modules
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.
Uniqueness of models in persistent homology: the case of curves
We consider generic curves in , i.e. generic C1 functions . We analyze these curves through the persistent homology groups of a filtration induced on S1 by f. In particular, we consider the question
Sketches of a platypus: persistent homology and its algebraic foundations
The various choices in use, and what they allow us to prove are examined, and the inherent differences between the choices people use are discussed, and potential directions of research are speculated on.
G‐invariant persistent homology
The stability of G-invariant persistent homology with respect to the natural pseudo-distance d_G is proved, and it is shown how this idea could be used in applications concerning shape comparison.


The theory of multidimensional persistence
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.
Multidimensional persistent homology is stable
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional
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
One-dimensional reduction of multidimensional persistent homology
A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a
Proximity of persistence modules and their diagrams
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes 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.
Introduction to Piecewise-Linear Topology
1. Polyhedra and P.L. Maps.- Basic Notation.- Joins and Cones.- Polyhedra.- Piecewise-Linear Maps.- The Standard Mistake.- P. L. Embeddings.- Manifolds.- Balls and Spheres.- The Poincare Conjecture
Describing shapes by geometrical-topological properties of real functions
This survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner.
Topology and data
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Survey of graph database models
The main objective of this survey is to present the work that has been conducted in the area of graph database modeling, concentrating on data structures, query languages, and integrity constraints.
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