Persistent homology is a method for computing topological features of a space at different spatial resolutions.
More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters.
A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets.
[2] A similar construction uses a nested sequence of Vietoris–Rips complexes known as the Vietoris–Rips filtration.
[3] Formally, consider a real-valued function on a simplicial complex
that is non-decreasing on increasing sequences of faces, so
(which is in practice always finite) induces an ordering on the sublevel complexes that defines a filtration When
on the simplicial homology groups for each dimension
persistent homology groups are the images of these homomorphisms, and the
coincide with the size function, a predecessor of persistent homology.
can be brought by a linear transformation preserving the filtration to so called canonical form, a canonically defined direct sum of filtered complexes of two types: one-dimensional complexes with trivial differential
and two-dimensional complexes with trivial homology
[6] A persistence module over a partially ordered set
There is a classification of persistence modules over a field
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
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,[6] where each generator is represented by a segment connecting the birth and the death values plotted on separate lines for each
Persistent homology is stable in a precise sense, which provides robustness against noise.
The bottleneck distance is a natural metric on the space of persistence diagrams given by
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
-metric on functions and the bottleneck distance on persistence diagrams.
[8] The principal algorithm is based on the bringing of the filtered complex to its canonical form by upper-triangular matrices and runs in worst-case cubical complexity in the number of simplices.
[6] The fastest known algorithm for computing persistent homology runs in matrix multiplication time.
[9] Since the number of simplices is highly relevant for computation time, finding filtered simplicial complexes with few simplexes is an active research area.
Several approaches have been proposed to reduce the number of simplices in a filtered simplicial complex in order to approximate persistent homology.
[10] [11] [12] [13] There are various software packages for computing persistence intervals of a finite filtration.