Computing Multidimensional Persistence

@article{Carlsson2009ComputingMP,
  title={Computing Multidimensional Persistence},
  author={Gunnar E. Carlsson and Gurjeet Singh and Afra Zomorodian},
  journal={ArXiv},
  year={2009},
  volume={abs/0907.2423}
}
The theory of multidimensional persistence captures the topology of a multifiltration --- a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence. 
Invariants for Multidimensional Persistence
The amount of data that our digital society collects is unprecedented. This represents a valuable opportunity to improve our quality of life by gaining insights about complex problems related to ne
Multidimensional Persistence and Noise
TLDR
The feature counting invariant is introduced and it is proved that assigning this invariant to compact tame functors is a 1-Lipschitz operation.
An Introduction to Multiparameter Persistence
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single
Necessary conditions for discontinuities of multidimensional persistent Betti numbers
Topological persistence has proven to be a promising framework for dealing with problems concerning the analysis of data. In this context, it was originally introduced by taking into account
Computing Multipersistence by Means of Spectral Systems
TLDR
This paper shows that spectral sequences and persistent homology are related, generalizing results valid in the case of filtrations over \Z, and develops a new module for the Kenzo system computing multipersistence.
A new approximation Algorithm for the Matching Distance in Multidimensional Persistence
TLDR
This paper proposes a new computational framework to deal with the multidimensional matching distance, by proving some new theoretical results and using them to formulate an algorithm for computing such a distance up to an arbitrary threshold error.
P-persistent homology of finite topological spaces
TLDR
It is shown that for any reasonable P-persistent object X in the category of finite topological spaces, there is a P− weighted graph, whose clique complex has the same P-Persistent homology as X.
Reducing complexes in multidimensional persistent homology theory
Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules
  • T. Dey, Cheng Xin
  • Computer Science
    Journal of Applied and Computational Topology
  • 2022
TLDR
A generalization of the persistence algorithm based on a generalized matrix reduction technique that runs in $O(n^{2\omega})$ time where $\omega<2.373$ is the exponent for matrix multiplication.
Morse-based Fibering of the Persistence Rank Invariant
TLDR
It is shown how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 34 REFERENCES
Computing Multidimensional Persistence
TLDR
A polynomial time algorithm for computing multidimensional persistence is given and this computation is recast as a problem within computational commutative algebra and utilize algorithms from this area to solve it.
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.
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.
Algebraic Topology
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.
Size homotopy groups for computation of natural size distances
For every manifold M endowed with a structure described by a function from M to the vector space R k , a parametric family of groups, called size homotopy groups, is introduced and studied. Some
An introduction to Morse theory
Morse theory on surfaces Extension to general dimensions Handelbodies Homology of manifolds Low dimensional manifolds A view from current mathematics Answers to exercises Bibliography Recommended
On hyperbolic groups
Abstract We prove that a δ-hyperbolic group for δ < ½ is a free product F * G 1 * … * Gn where F is a free group of finite rank and each Gi is a finite group.
Computing persistent homology
TLDR
The homology of a filtered d-dimensional simplicial complex K is studied as a single algebraic entity and a correspondence is established that provides a simple description over fields that enables a natural algorithm for computing persistent homology over an arbitrary field in any dimension.
Using Algebraic Geometry
TLDR
The Berlekamp-Massey-Sakata Decoding Algorithm is used for solving Polynomial Equations and for computations in Local Rings.
Vines and vineyards by updating persistence in linear time
TLDR
The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering and uses the algorithm to compute 1-parameter families of diagrams which are applied to the study of protein folding trajectories.
...
1
2
3
4
...