Persistent Homology Rik Sweep, 0850929 Eindhoven, university of - PowerPoint PPT Presentation

Persistent Homology Rik Sweep, 0850929 Eindhoven, university of Technology 17 th May 2018

Subjects ◮ Motivation ◮ Filtrations ◮ Persistence ◮ Barcodes ◮ fundamental Theorem

Motivation X

Motivation → X P

Motivation ? ← X P

Motivation → ˇ C 2 ( P ) P

Motivation → ˇ P C 4 ( P )

Motivation → P ˇ C 68 ( P )

Motivation Which radius to choose?

Motivation Which radius to choose? Persistent homology can help us!

Motivation Which radius to choose? Persistent homology can help us! Look at how the (ˇ Cech-) complexes grow.

Filtration A Filtration is a sequence of simplicial complexes contained in each other. K 1 ⊆ K 2 ⊆ · · · ⊆ K N

Filtration A Filtration K is a sequence of simplicial complexes contained in each other. K : K 1 ⊆ K 2 ⊆ · · · ⊆ K N Example:

Filtration ◮ How to extract information from such a filtration?

Filtration ◮ How to extract information from such a filtration? ◮ What happens for one complex?

Homology Groups ◮ Distinguish shapes by examining holes.

Homology Groups ◮ Distinguish shapes by examining holes. ◮ Recall the formal definition: H p ( K i ) = Z p / B p = Ker( ∂ p ) / Im( ∂ p +1 )

Homology Groups ◮ Distinguish shapes by examining holes. ◮ Recall the formal definition: H p ( K i ) = Z p / B p = Ker( ∂ p ) / Im( ∂ p +1 ) ◮ How to use this when you have many complexes K i ?

Maps f ij p The maps f ij p map p -dimensional holes of a (ˇ Cech-) complex K i to ”the same” holes in the (ˇ Cech-) complex K j . f ij p : H p ( K i ) → H p ( K j )

Maps f ij p Example of such a f ij p P

Maps f ij p Example of such a f ij p ˇ Cech complex with radius ε 1 = 3

Maps f ij p Example of such a f ij p ˇ Cech complex with radius ε 2 = 5

Maps f ij p Example of such a f ij p Both ˇ Cech complexes.

Maps f ij p Example of such a f ij p Both ˇ Cech complexes. Notice how this is (part of) a filtration!

Persistent Homology Groups ◮ The p th persistent homology group between K i and K j , i < j , is defined as the following vector space. H i , j p ( K ) = Im( f ij p )

Persistent Homology Groups ◮ The p th persistent homology group between K i and K j , i < j , is defined as the following vector space. H i , j p ( K ) = Im( f ij p ) ◮ So it is the group of p -dimensional holes that existed in K i and still exist in K j

Persistent Homology Groups Recall the previous example. ◮ So it is the group of p -dimensional holes that existed in K i and still exist in K j

Persistent Betti Numbers ◮ Each p th persistent homology group H i , j p ( K ) has its corresponding persistent Betti number β i , j p that is defined as follows. β i , j p = dim H i , j p ( K )

Persistent Betti Numbers ◮ Each p th persistent homology group H i , j p ( K ) has its corresponding persistent Betti number β i , j p that is defined as follows. β i , j p = dim H i , j p ( K ) ◮ So the p th persistent Betti number is the number of holes that existed in K i and still exist in K j .

Persistent Betti Numbers ◮ What do we do with these Persistent Betti numbers?

Persistent Betti Numbers ◮ What do we do with these Persistent Betti numbers? ◮ We can use them to define so-called barcodes.

Barcodes ◮ The p th barcode is a multiset (a set with multiplicities) of intervals [ i , j ) with 0 ≤ i < j ≤ N such that each interval [ i , j ) has multiplicity µ i , j p where µ i , j p = β i , j − 1 − β i − 1 , j − 1 − β i , j p + β i − 1 , j p p p and intervals [ i , ∞ ) with multiplicity µ i , ∞ where p µ i , ∞ = β i , N − β i − 1 , N p p p

Barcodes ◮ Let us construct the 0 th barcode of the previous example. K 0 K 1

Barcodes ◮ Let us construct the 0 th barcode of the previous example. K 0 K 1 ◮ Recall µ i , j p = β i , j − 1 − β i − 1 , j − 1 − β i , j p + β i − 1 , j p p p µ i , ∞ = β i , N − β i − 1 , N p p p

Barcodes ◮ Let us construct the 0 th barcode of the previous example. K 0 K 1 intervals 0 3 5

Barcodes ◮ Let us construct the 1 st barcode of the previous example. K 0 K 1 ◮ Recall µ i , j p = β i , j − 1 − β i − 1 , j − 1 − β i , j p + β i − 1 , j p p p µ i , ∞ = β i , N − β i − 1 , N p p p

Barcodes ◮ Let us construct the 1 st barcode of the previous example. K 0 K 1 intervals 0 3 5

Barcodes ◮ Let’s look at a more elaborate example.

Barcodes ◮ Let’s look at a more elaborate example. r 1 = 1 r 2 = 2 r 3 = 3 r 4 = 4 r 5 = 5 r 6 = 44 r 7 = 65 r 8 = 68

Barcodes ◮ Let’s look at a more elaborate example. ◮ The 1 st barcode looks as follows.

Barcodes ◮ What is the use of these Barcodes?

Fundamental Theorem of Persistent Homology ◮ Links barcodes to the persistent homology groups.

Fundamental Theorem of Persistent Homology ◮ Links barcodes to persistent homology groups. For every p ≥ 0, the p th barcode of K is well defined and for all 0 ≤ i ≤ j ≤ N , the dimension of the persistent homology group p ( K ) equals the number of intervals in the p th barcode of K H i , j (counted with multiplicities) which contain the interval [ i , j ]. In particular, dim H p ( K i ) equals the number of intervals in the p th barcode of K (counted with multiplicities) which contain i .

Proof as the number of intervals in the p th barcode that Define I i , j p contain [ i , j ]. I i , j = p . . . = dim H i , j p

Proof as the number of intervals in the p th barcode that Define I i , j p contain [ i , j ]. I i , j = p

Proof as the number of intervals in the p th barcode that Define I i , j p contain [ i , j ]. � � I i , j µ k , l = p p k ≤ i l ≥ j +1

Proof as the number of intervals in the p th barcode that Define I i , j p contain [ i , j ]. � � I i , j µ k , l = p p k ≤ i l ≥ j +1 � � � � � � β k , l − 1 − β k , l β k − 1 , l − 1 − β k − 1 , l = − p p p p k ≤ i l ≥ j +1

Proof as the number of intervals in the p th barcode that Define I i , j p contain [ i , j ]. � � I i , j µ k , l = p p k ≤ i l ≥ j +1 � � � � � � β k , l − 1 − β k , l β k − 1 , l − 1 − β k − 1 , l = − p p p p k ≤ i l ≥ j +1 � β k , j − β k − 1 , j +1 = p p k ≤ i

Proof as the number of intervals in the p th barcode that Define I i , j p contain [ i , j ]. � � I i , j µ k , l = p p k ≤ i l ≥ j +1 � � � � � � β k , l − 1 − β k , l β k − 1 , l − 1 − β k − 1 , l = − p p p p k ≤ i l ≥ j +1 � β k , j − β k − 1 , j = p p k ≤ i = β i , j p

Proof as the number of intervals in the p th barcode that Define I i , j p contain [ i , j ]. I i , j � � µ k , l = p p k ≤ i l ≥ j +1 � � � � � � β k , l − 1 − β k , l β k − 1 , l − 1 − β k − 1 , l = − p p p p k ≤ i l ≥ j +1 � β k , j − β k − 1 , j = p p k ≤ i = β i , j p = dim H i , j p

Conclusion ◮ If you don’t know what radius you need, try multiple

Conclusion ◮ If you don’t know what parameter value you need, try multiple. ◮ Once you have chosen multiple values, use persistent homology to investigate the shape.

Conclusion ◮ If you don’t know what parameter value you need, try multiple. ◮ Once you have chosen multiple values, use persistent homology to investigate the shape. ◮ Visualize the holes in the shape using barcodes.

References ◮ Francisco Belchi, Aniceto Murillo, ” A ∞ -persistence”, Applicable Algebra in Engineering, Communication and Computing: Volume 26, Issue 1 (2015), pp 121-139 ◮ Afra Zomorodian, Gunnar Carlsson, ”Computing Persistent Homology”, Discrete and Computational Geometry: Volume 33, Issue 2 (2005), pp 249-274