1. Persistent Homology
the basics
Image:Jean-Marie Hullot (CC BY 3.0)
kelly davis (founder forty.to)

2. problem:what’s the topology of a point cloud?
Image:Jean-Marie Hullot (CC BY 3.0)

3. solution: persistent homology
Image:Jean-Marie Hullot (CC BY 2.0)

4. simplicial complex
Image:Antoine Hubert (CC BY 2.0)

5. simplicial homology
Image:Jean-Marie Hullot (CC BY 2.0)

6. simplicial homology
i
K L
Image:Jean-Marie Hullot (CC BY 2.0)

7. simplicial filtrations
iK Ki
Image:Jean-Marie Hullot (CC BY 3.0)

8. birth, death, and taxes
it is born at Ki if γ ∈ Hp . Furthermore, if γ is born at Ki then it dies
entering Kj if it merges with an older class as we go from Kj−1 to Kj, that is,
fi,j−1
p (γ) ∈ Hi−1,j−1
p but fi,j
p (γ) ∈ Hi−1,j
p ; see Figure VII.2. This is again the
0 0 0 0
H p
i−1
H
i
H p
j −1
H pp
j
γ
Figure VII.2: The class γ is born at Ki since it does not lie in the (shaded) image
of Hi−1
p . Furthermore, γ dies entering Kj since this is the first time its image merges
into the image of Hi−1
p .
Elder Rule. If γ is born at Ki and dies entering Kj then we call the difference
in function value the persistence, pers(γ) = aj − ai. Sometimes we prefer to
ignore 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. persistence
it is born at Ki if γ ∈ Hp . Furthermore, if γ is born at Ki then it dies
entering Kj if it merges with an older class as we go from Kj−1 to Kj, that is,
fi,j−1
p (γ) ∈ Hi−1,j−1
p but fi,j
p (γ) ∈ Hi−1,j
p ; see Figure VII.2. This is again the
0 0 0 0
H p
i−1
H
i
H p
j −1
H pp
j
γ
Figure VII.2: The class γ is born at Ki since it does not lie in the (shaded) image
of Hi−1
p . Furthermore, γ dies entering Kj since this is the first time its image merges
into the image of Hi−1
p .
Elder Rule. If γ is born at Ki and dies entering Kj then we call the difference
in function value the persistence, pers(γ) = aj − ai. Sometimes we prefer to
ignore 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. so what!
2.3. Barcodes. The parameter intervals arising from the basis for H∗(C; F) in
Equation (2.3) inspire a visual snapshot of Hk(C; F) in the form of a barcode. A
barcode is a graphical representation of Hk(C; F) as a collection of horizontal line
segments in a plane whose horizontal axis corresponds to the parameter and whose
vertical axis represents an (arbitrary) ordering of homology generators. Figure 4
gives an example of barcode representations of the homology of the sampling of
points in an annulus from Figure 3 (illustrated in the case of a large number of
parameter values i).
H0
H1
H2
Figure 4. [bottom] An example of the barcodes for H∗(R) in the
example of Figure 3. [top] The rank of Hk(R i
) equals the number
of 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→j
k (C; F) is
equal to the number of intervals in the barcode of Hk(C; F) spanning the parameter
i Image:Padmanaba01 (CC BY 2.0)

11. calm before the algorithm
Image:Pierre-Emmanuel BOITON (CC BY 2.0)

12. the algorithm: betti numbers
Image:Pierre-Emmanuel BOITON (CC BY 2.0)

13. the algorithm: persistence
Image:Pierre-Emmanuel BOITON (CC BY 2.0)

14. implementations:
phat - c++ with c++ api
dionysus - c++ with python api
plex - java with java api
Image:Jean-Marie Hullot (CC BY 3.0)
Persistent Homology
the basics
kelly davis (founder forty.to)