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Persistent homology

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Persistent homology

1. 1. Persistent Homologythe basicsImage:Jean-Marie Hullot (CC BY 3.0)kelly davis (founder forty.to)
2. 2. problem:what’s the topology of a point cloud?Image:Jean-Marie Hullot (CC BY 3.0)
3. 3. solution: persistent homologyImage:Jean-Marie Hullot (CC BY 2.0)
4. 4. simplicial complexImage:Antoine Hubert (CC BY 2.0)
5. 5. simplicial homologyImage:Jean-Marie Hullot (CC BY 2.0)
6. 6. simplicial homologyiK LImage:Jean-Marie Hullot (CC BY 2.0)
7. 7. simplicial ﬁltrationsiK KiImage:Jean-Marie Hullot (CC BY 3.0)
8. 8. birth, death, and taxesit is born at Ki if γ ∈ Hp . Furthermore, if γ is born at Ki then it diesentering Kj if it merges with an older class as we go from Kj−1 to Kj, that is,fi,j−1p (γ) ∈ Hi−1,j−1p but fi,jp (γ) ∈ Hi−1,jp ; see Figure VII.2. This is again the0 0 0 0H pi−1HiH pj −1H ppjγFigure VII.2: The class γ is born at Ki since it does not lie in the (shaded) imageof Hi−1p . Furthermore, γ dies entering Kj since this is the ﬁrst time its image mergesinto the image of Hi−1p .Elder Rule. If γ is born at Ki and dies entering Kj then we call the diﬀerencein function value the persistence, pers(γ) = aj − ai. Sometimes we prefer toignore the actual function values and consider the diﬀerence in index, j − i,which we call the index persistence of the class. If γ is born at Ki but neverImage:Jean-Marie Hullot (CC BY 3.0)
9. 9. persistenceit is born at Ki if γ ∈ Hp . Furthermore, if γ is born at Ki then it diesentering Kj if it merges with an older class as we go from Kj−1 to Kj, that is,fi,j−1p (γ) ∈ Hi−1,j−1p but fi,jp (γ) ∈ Hi−1,jp ; see Figure VII.2. This is again the0 0 0 0H pi−1HiH pj −1H ppjγFigure VII.2: The class γ is born at Ki since it does not lie in the (shaded) imageof Hi−1p . Furthermore, γ dies entering Kj since this is the ﬁrst time its image mergesinto the image of Hi−1p .Elder Rule. If γ is born at Ki and dies entering Kj then we call the diﬀerencein function value the persistence, pers(γ) = aj − ai. Sometimes we prefer toignore the actual function values and consider the diﬀerence in index, j − i,which we call the index persistence of the class. If γ is born at Ki but neverImage:Jean-Marie Hullot (CC BY 3.0)
10. 10. so what!2.3. Barcodes. The parameter intervals arising from the basis for H∗(C; F) inEquation (2.3) inspire a visual snapshot of Hk(C; F) in the form of a barcode. Abarcode is a graphical representation of Hk(C; F) as a collection of horizontal linesegments in a plane whose horizontal axis corresponds to the parameter and whosevertical axis represents an (arbitrary) ordering of homology generators. Figure 4gives an example of barcode representations of the homology of the sampling ofpoints in an annulus from Figure 3 (illustrated in the case of a large number ofparameter values i).H0H1H2Figure 4. [bottom] An example of the barcodes for H∗(R) in theexample of Figure 3. [top] The rank of Hk(R i) equals the numberof intervals in the barcode for Hk(R) intersecting the (dashed) line= i.Theorem 2.3 yields the fundamental characterization of barcodes.Theorem 2.4 ([22]). The rank of the persistent homology group Hi→jk (C; F) isequal to the number of intervals in the barcode of Hk(C; F) spanning the parameteri Image:Padmanaba01 (CC BY 2.0)
11. 11. calm before the algorithmImage:Pierre-Emmanuel BOITON (CC BY 2.0)
12. 12. the algorithm: betti numbersImage:Pierre-Emmanuel BOITON (CC BY 2.0)
13. 13. the algorithm: persistenceImage:Pierre-Emmanuel BOITON (CC BY 2.0)
14. 14. implementations:phat - c++ with c++ apidionysus - c++ with python apiplex - java with java apiImage:Jean-Marie Hullot (CC BY 3.0)Persistent Homologythe basicskelly davis (founder forty.to)