“Persistent homology”的意思、由来-开放百科全书

To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets.

Definition

Formally, consider a real-valued function on a simplicial complex

that is non-decreasing on increasing sequences of faces, so

whenever

is a face of

. Then for every

the sublevel set

is a subcomplex of K, and the ordering of the values of

on the simplices in

(which is in practice always finite) induces an ordering on the sublevel complexes that defines a filtration

When

, the inclusion

induces a homomorphism

on the simplicial homology groups for each dimension

. The

persistent homology groups are the images of these homomorphisms, and the

persistent Betti numbers

are the ranks of those groups.^[2] Persistent Betti numbers for

coincide with

A persistence module over a partially ordered set

is a set of vector spaces

indexed by

, with a linear map

whenever

, with

equal to the identity and

for

. Equivalently, we may consider it as a functor from

considered as a category to the category of vector spaces (or

-modules). There is a classification of persistence modules over a field

indexed by

Multiplication by

corresponds to moving forward one step in the persistence module. Intuitively, the free parts on the right side correspond to the homology generators that appear at filtration level

and never disappear, while the torsion parts correspond to those that appear at filtration level

and last for

steps of the filtration (or equivalently, disappear at filtration level

).^[4] ^[5]

This theorem allows us to uniquely represent the persistent homology of a filtered simplicial complex with a barcode or persistence diagram. A barcode represents each persistent generator with a horizontal line beginning at the first filtration level where it appears, and ending at the filtration level where it disappears, while a persistence diagram plots a point for each generator with its x-coordinate the birth time and its y-coordinate the death time.

Equivalently the same data is represented by Barannikov's canonical form^[5], where each generator is represented by a segment connecting the birth and the death values plotted on separate lines for each

Stability

Persistent homology is stable in a precise sense, which provides robustness against noise. There is a natural metric on the space of persistence diagrams given by

called the bottleneck distance. A small perturbation in the input filtration leads to a small perturbation of its persistence diagram in the bottleneck distance. For concreteness, consider a filtration on a space

homeomorphic to a simplicial complex determined by the sublevel sets of a continuous tame function

. The map

taking

to the persistence diagram of its

th homology is 1-Lipschitz with respect to the

-metric on functions and the bottleneck distance on persistence diagrams.

Computation

There are various software packages for computing persistence intervals of a finite filtration. The principal algorithm is based on the bringing of the filtered complex to its canonical form by upper-triangular matrices^[5].

See also

References

1. ^Carlsson, Gunnar (2009). "Topology and data". AMS Bulletin 46(2), 255–308.
2. ^Edelsbrunner, H and Harer, J (2010). [https://books.google.com/books?id=MDXa6gFRZuIC&printsec=frontcover#v=onepage&q=%22persistent%20homology%22&f=false Computational Topology: An Introduction]. American Mathematical Society.
3. ^Verri, A, Uras, C, Frosini, P and Ferri, M (1993). [https://link.springer.com/article/10.1007/BF00200823#page-1 On the use of size functions for shape analysis], Biological Cybernetics, 70, 99–107.
4. ^{{Cite journal|title = Computing Persistent Homology|journal = Discrete & Computational Geometry|date = 2004-11-19|issn = 0179-5376|pages = 249–274|volume = 33|issue = 2|doi = 10.1007/s00454-004-1146-y|first = Afra|last = Zomorodian|first2 = Gunnar|last2 = Carlsson}}
5. ^¹²{{Cite journal|title = Framed Morse complex and its invariants |url = https://www.researchgate.net/publication/267672645 |journal = Advances in Soviet Mathematics | date = 1994|pages = 93–115|volume = 21|first = Sergey|last = Barannikov}}
6. ^{{Cite journal|title = Stability of Persistence Diagrams|journal = Discrete & Computational Geometry|date = 2006-12-12|issn = 0179-5376|pages = 103–120|volume = 37|issue = 1|doi = 10.1007/s00454-006-1276-5|first = David|last = Cohen-Steiner|first2 = Herbert|last2 = Edelsbrunner|first3 = John|last3 = Harer}}
7. ^¹Licenses here are a summary, and are not taken to be complete statements of the licenses. Some packages may use libraries under different licenses.