The Theory of Multidimensional Persistence

@article{Carlsson2007TheTO,
  title={The Theory of Multidimensional Persistence},
  author={Gunnar E. Carlsson and Afra Zomorodian},
  journal={Discrete \& Computational Geometry},
  year={2007},
  volume={42},
  pages={71-93}
}
Persistent homology captures the topology of a filtration—a one-parameter family of increasing spaces—in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional… 
View on ACM
link.springer.com
388 Citations
Highly Influential Citations
49
Background Citations
208
Methods Citations
50
Results Citations
3

Figures from this paper

388 Citations

Multiparameter persistent homology via generalized Morse theory
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that…
On the complexity of zero-dimensional multiparameter persistence
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full…
The Persistence Space in Multidimensional Persistent Homology
TLDR
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.
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.
A Study of Monodromy in the Computation of Multidimensional Persistence
TLDR
This contribution has identified instances of nontrivial monodromy over loops in nonsingular parameter space and given an example of a filtration in which it can be shown to be nontrivials.
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.
A refinement of multi-dimensional persistence
We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on…
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…
Estimating Multidimensional Persistent Homology Through a Finite Sampling
TLDR
It is shown that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of X from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain.
Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension
TLDR
This work proposes a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size and provides realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.
...
...

References

SHOWING 1-10 OF 38 REFERENCES
A refinement of multi-dimensional persistence
We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on…
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.
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.
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.
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.
Geometric Invariant Theory
“Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory to…
Gauss-Manin connections for arrangements, IV Nonresonant eigenvalues
An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for…
The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry
Preface: Algebra and Geometry * Free Resolutions and Hilbert Functions * First Examples of Free Resolutions * Points in P^2 * Castelnuovo-Mumford Regularity * The Regularity of Projective Curves *…
Representations of Finite-Dimensional Algebras
1. Matrix Problems.- 2. Algebras, Modules and Categories.- 3. Radical, Decomposition, Aggregates.- 4. Finitely Spaced Modules.- 5. Finitely Represented Posets.- 6. Roots.- 7. Representations of…
Moduli space of combinatorially equivalent arrangements of hyperplanes and logarithmic Gauss–Manin connections
...
...