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Persistent Homology Rik Sweep, 0850929 Eindhoven, university of - - PowerPoint PPT Presentation
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
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Motivation
X
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Motivation
X
→
P
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Motivation
X
?
←
P
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Motivation
P
→
ˇ C2(P)
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Motivation
P
→
ˇ C4(P)
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Motivation
P
→
ˇ C68(P)
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Motivation
Which radius to choose?
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Motivation
Which radius to choose? Persistent homology can help us!
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Motivation
Which radius to choose? Persistent homology can help us! Look at how the (ˇ Cech-) complexes grow.
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Filtration
A Filtration is a sequence of simplicial complexes contained in each
- ther.
K1 ⊆ K2 ⊆ · · · ⊆ KN
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Filtration
A Filtration K is a sequence of simplicial complexes contained in each other. K : K1 ⊆ K2 ⊆ · · · ⊆ KN Example:
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Filtration
◮ How to extract information from such a filtration?
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Filtration
◮ How to extract information from such a filtration? ◮ What happens for one complex?
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Homology Groups
◮ Distinguish shapes by examining holes.
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Homology Groups
◮ Distinguish shapes by examining holes. ◮ Recall the formal definition:
Hp(Ki) = Zp/Bp = Ker(∂p)/Im(∂p+1)
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Homology Groups
◮ Distinguish shapes by examining holes. ◮ Recall the formal definition:
Hp(Ki) = Zp/Bp = Ker(∂p)/Im(∂p+1)
◮ How to use this when you have many complexes Ki?
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Maps f ij
p
The maps f ij
p map p-dimensional holes of a (ˇ
Cech-) complex Ki to ”the same” holes in the (ˇ Cech-) complex Kj. f ij
p : Hp(Ki) → Hp(Kj)
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Maps f ij
p
Example of such a f ij
p
P
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Maps f ij
p
Example of such a f ij
p
ˇ Cech complex with radius ε1 = 3
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Maps f ij
p
Example of such a f ij
p
ˇ Cech complex with radius ε2 = 5
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Maps f ij
p
Example of such a f ij
p
Both ˇ Cech complexes.
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Maps f ij
p
Example of such a f ij
p
Both ˇ Cech complexes.
Notice how this is (part of) a filtration!
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Persistent Homology Groups
◮ The pth persistent homology group between Ki and Kj, i < j,
is defined as the following vector space. Hi,j
p (K) = Im(f ij p )
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Persistent Homology Groups
◮ The pth persistent homology group between Ki and Kj, i < j,
is defined as the following vector space. Hi,j
p (K) = Im(f ij p ) ◮ So it is the group of p-dimensional holes that existed in Ki
and still exist in Kj
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Persistent Homology Groups
Recall the previous example.
◮ So it is the group of p-dimensional holes that existed in Ki
and still exist in Kj
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Persistent Betti Numbers
◮ Each pth persistent homology group Hi,j p (K) has its
corresponding persistent Betti number βi,j
p that is defined as
follows. βi,j
p = dim Hi,j p (K)
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Persistent Betti Numbers
◮ Each pth persistent homology group Hi,j p (K) has its
corresponding persistent Betti number βi,j
p that is defined as
follows. βi,j
p = dim Hi,j p (K) ◮ So the pth persistent Betti number is the number of holes
that existed in Ki and still exist in Kj.
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Persistent Betti Numbers
◮ What do we do with these Persistent Betti numbers?
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Persistent Betti Numbers
◮ What do we do with these Persistent Betti numbers? ◮ We can use them to define so-called barcodes.
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Barcodes
◮ The pth 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 µi,j
p where
µi,j
p = βi,j−1 p
− βi−1,j−1
p
− βi,j
p + βi−1,j p
and intervals [i, ∞) with multiplicity µi,∞
p
where µi,∞
p
= βi,N
p
− βi−1,N
p
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Barcodes
◮ Let us construct the 0th barcode of the previous example.
K0 K1
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Barcodes
◮ Let us construct the 0th barcode of the previous example.
K0 K1
◮ Recall
µi,j
p = βi,j−1 p
− βi−1,j−1
p
− βi,j
p + βi−1,j p
µi,∞
p
= βi,N
p
− βi−1,N
p
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Barcodes
◮ Let us construct the 0th barcode of the previous example.
K0 K1
intervals
3 5
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Barcodes
◮ Let us construct the 1st barcode of the previous example.
K0 K1
◮ Recall
µi,j
p = βi,j−1 p
− βi−1,j−1
p
− βi,j
p + βi−1,j p
µi,∞
p
= βi,N
p
− βi−1,N
p
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Barcodes
◮ Let us construct the 1st barcode of the previous example.
K0 K1
intervals
3 5
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Barcodes
◮ Let’s look at a more elaborate example.
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Barcodes
◮ Let’s look at a more elaborate example.
r1 = 1 r2 = 2 r3 = 3 r4 = 4 r5 = 5 r6 = 44 r7 = 65 r8 = 68
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Barcodes
◮ Let’s look at a more elaborate example. ◮ The 1st barcode looks as follows.
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Barcodes
◮ What is the use of these Barcodes?
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Fundamental Theorem of Persistent Homology
◮ Links barcodes to the persistent homology groups.
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Fundamental Theorem of Persistent Homology
◮ Links barcodes to persistent homology groups.
For every p ≥ 0, the pth barcode of K is well defined and for all 0 ≤ i ≤ j ≤ N, the dimension of the persistent homology group Hi,j
p (K) equals the number of intervals in the pth barcode of K
(counted with multiplicities) which contain the interval [i, j]. In particular, dim Hp(Ki) equals the number of intervals in the pth barcode of K (counted with multiplicities) which contain i.
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Proof
Define I i,j
p
as the number of intervals in the pth barcode that contain [i, j]. I i,j
p
= . . . = dim Hi,j
p
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Proof
Define I i,j
p
as the number of intervals in the pth barcode that contain [i, j]. I i,j
p
=
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Proof
Define I i,j
p
as the number of intervals in the pth barcode that contain [i, j]. I i,j
p
=
- k≤i
- l≥j+1
µk,l
p
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Proof
Define I i,j
p
as the number of intervals in the pth barcode that contain [i, j]. I i,j
p
=
- k≤i
- l≥j+1
µk,l
p
=
- k≤i
- l≥j+1
- βk,l−1
p
− βk,l
p
- −
- βk−1,l−1
p
− βk−1,l
p
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Proof
Define I i,j
p
as the number of intervals in the pth barcode that contain [i, j]. I i,j
p
=
- k≤i
- l≥j+1
µk,l
p
=
- k≤i
- l≥j+1
- βk,l−1
p
− βk,l
p
- −
- βk−1,l−1
p
− βk−1,l
p
- =
- k≤i
βk,j
p
− βk−1,j+1
p
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Proof
Define I i,j
p
as the number of intervals in the pth barcode that contain [i, j]. I i,j
p
=
- k≤i
- l≥j+1
µk,l
p
=
- k≤i
- l≥j+1
- βk,l−1
p
− βk,l
p
- −
- βk−1,l−1
p
− βk−1,l
p
- =
- k≤i
βk,j
p
− βk−1,j
p
= βi,j
p
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Proof
Define I i,j
p
as the number of intervals in the pth barcode that contain [i, j]. I i,j
p
=
- k≤i
- l≥j+1
µk,l
p
=
- k≤i
- l≥j+1
- βk,l−1
p
− βk,l
p
- −
- βk−1,l−1
p
− βk−1,l
p
- =
- k≤i
βk,j
p
− βk−1,j
p
= βi,j
p
= dim Hi,j
p
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Conclusion
◮ If you don’t know what radius you need, try multiple
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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.
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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.
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