Puneet Varma (Editor)

Persistent homology

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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 length and are deemed more likely to represent true features of the underlying space, rather than artifacts of sampling, noise, or particular choice of parameters.

Contents

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 f : K → R that is non-decreasing on increasing sequences of faces, so f ( σ ) ≤ f ( τ ) whenever σ is a face of τ in K . Then for every a ∈ R the sublevel set K ( a ) = f − 1 ( − ∞ , a ] is a subcomplex of K, and the ordering of the values of f on the simplices in K (which is in practice always finite) induces an ordering on the sublevel complexes that defines a filtration

∅ = K 0 ⊆ K 1 ⊆ … ⊆ K n = K

When 0 ≤ i ≤ j ≤ n , the inclusion K i ↪ K j induces a homomorphism f p i , j : H p ( K i ) → H p ( K j ) on the simplicial homology groups for each dimension p . The p t h persistent homology groups are the images of these homomorphisms, and the p t h persistent Betti numbers β p i , j are the ranks of those groups. Persistent Betti numbers for p = 0 coincide with the size function, a predecessor of persistent homology.

A persistence module over a partially ordered set P is a set of vector spaces U t indexed by P , with a linear map u t s : U s → U t whenever s ≤ t , with u t t equal to the identity and u t s ∘ u s r = u t r for r ≤ s ≤ t . Equivalently, we may consider it as a functor from P considered as a category to the category of vector spaces (or R -modules). There is a classification of persistence modules over a field F indexed by N :

x t i r j s j s j + r j

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.

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

bottleneck distance X f : X → R f k sup W ∞ ( D ( f ) , D ( g ) ) ≤ ∥ f − g ∥ ∞

Computation

There are various software packages for computing persistence intervals of a finite filtration.

References

Persistent homology Wikipedia
(Text) CC BY-SA


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