persistent homology

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

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problem:what’s the topology of a point cloud?

Image:Jean-Marie Hullot (CC BY 3.0)

solution: persistent homology

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simplicial complex

Image:Antoine Hubert (CC BY 2.0)

simplicial homology

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simplicial homology

i

K L

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simplicial filtrations

iK Ki

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birth, death, and taxes

180 VII Persistence

each other. We collect the classes that are born at or before a given thresholdand die after another threshold in groups.

Definition. The p-th persistent homology groups are the images of thehomomorphisms induced by inclusion, Hi,j

p = im f i,jp , for 0 ! i ! j ! n. The

corresponding p-th persistent Betti numbers are the ranks of these groups,!i,j

p = rankHi,jp .

Similarly, we define reduced persistent homology groups and reduced persistentBetti numbers. Note that Hi,i

p = Hp(Ki). The persistent homology groupsconsist of the homology classes of Ki that are still alive at Kj or, more formally,Hi,j

p = Zp(Ki)/(Bp(Kj) " Zp(Ki)). We have such a group for each dimension pand each index pair i ! j. We can be more concrete about the classes countedby the persistent homology groups. Letting " be a class in Hp(Ki), we sayit is born at Ki if " #$ Hi!1,i

p . Furthermore, if " 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,f i,j!1

p (") #$ Hi!1,j!1p but f i,j

p (") $ Hi!1,jp ; see Figure VII.2. This is again the

0 0 0 0

H pi−1 H i H p

j −1 H ppj

γ

Figure VII.2: The class ! is born at Ki since it does not lie in the (shaded) imageof Hi!1

p . Furthermore, ! dies entering Kj since this is the first time its image mergesinto the image of Hi!1

p .

Elder Rule. If " is born at Ki and dies entering Kj then we call the di!erencein function value the persistence, pers(") = aj % ai. Sometimes we prefer toignore the actual function values and consider the di!erence in index, j % i,which we call the index persistence of the class. If " is born at Ki but neverdies then we set its persistence as well as its index persistence to infinity.

We note that births and deaths can also be defined for a sequence of vectorspaces that are not necessarily homology groups. All we need is a finite sequenceand homomorphisms from left to right which, for vector spaces, are usuallyreferred to as linear maps.

Image:Jean-Marie Hullot (CC BY 3.0)

persistence

180 VII Persistence

each other. We collect the classes that are born at or before a given thresholdand die after another threshold in groups.

Definition. The p-th persistent homology groups are the images of thehomomorphisms induced by inclusion, Hi,j

p = im f i,jp , for 0 ! i ! j ! n. The

corresponding p-th persistent Betti numbers are the ranks of these groups,!i,j

p = rankHi,jp .

Similarly, we define reduced persistent homology groups and reduced persistentBetti numbers. Note that Hi,i

p = Hp(Ki). The persistent homology groupsconsist of the homology classes of Ki that are still alive at Kj or, more formally,Hi,j

p = Zp(Ki)/(Bp(Kj) " Zp(Ki)). We have such a group for each dimension pand each index pair i ! j. We can be more concrete about the classes countedby the persistent homology groups. Letting " be a class in Hp(Ki), we sayit is born at Ki if " #$ Hi!1,i

p . Furthermore, if " 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,f i,j!1

p (") #$ Hi!1,j!1p but f i,j

p (") $ Hi!1,jp ; see Figure VII.2. This is again the

0 0 0 0

H pi−1 H i H p

j −1 H ppj

γ

Figure VII.2: The class ! is born at Ki since it does not lie in the (shaded) imageof Hi!1

p . Furthermore, ! dies entering Kj since this is the first time its image mergesinto the image of Hi!1

p .

Elder Rule. If " is born at Ki and dies entering Kj then we call the di!erencein function value the persistence, pers(") = aj % ai. Sometimes we prefer toignore the actual function values and consider the di!erence in index, j % i,which we call the index persistence of the class. If " is born at Ki but neverdies then we set its persistence as well as its index persistence to infinity.

We note that births and deaths can also be defined for a sequence of vectorspaces that are not necessarily homology groups. All we need is a finite sequenceand homomorphisms from left to right which, for vector spaces, are usuallyreferred to as linear maps.

Image:Jean-Marie Hullot (CC BY 3.0)

so what!

PERSISTENT TOPOLOGY OF DATA 67

which come into existence at parameter ti and which persist for all future parame-ter values. The torsional elements correspond to those homology generators whichappear at parameter rj and disappear at parameter rj + sj . At the chain level,the Structure Theorem provides a birth-death pairing of generators of C (exceptingthose that persist to infinity).

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).

H0

H1

H2!

!

!

Figure 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 ) is

equal to the number of intervals in the barcode of Hk(C; F ) spanning the parameterinterval [i, j]. In particular, H!(Ci

!; F ) is equal to the number of intervals whichcontain i.

A barcode is best thought of as the persistence analogue of a Betti number.Recall that the kth Betti number of a complex, "k := rank(Hk), acts as a coarsenumerical measure of Hk. As with "k, the barcode for Hk does not give any in-formation about the finer structure of the homology, but merely a continuously

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calm before the algorithm

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the algorithm: betti numbers

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the algorithm: persistence

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