Review. Basic Definitions. Example. Computational Algebraic Topology. Robert
Hank. Department of Mathematics. University of Minnesota. Junior Colloquium ...
Review Basic Definitions Example
Computational Algebraic Topology Robert Hank Department of Mathematics University of Minnesota
Junior Colloquium, 04/09/2012
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces Homotopy groups Homology groups Cohomology groups Cohomology ring structure Other homology and cohomology theories (SHT, K-theory)
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces Homotopy groups Homology groups Cohomology groups Cohomology ring structure Other homology and cohomology theories (SHT, K-theory)
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces Homotopy groups Homology groups Cohomology groups Cohomology ring structure Other homology and cohomology theories (SHT, K-theory)
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces Homotopy groups Homology groups Cohomology groups Cohomology ring structure Other homology and cohomology theories (SHT, K-theory)
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces Homotopy groups Homology groups Cohomology groups Cohomology ring structure Other homology and cohomology theories (SHT, K-theory)
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
We Prefer Algebra over Topology
Use algebra to distinguish spaces Homotopy groups Homology groups Cohomology groups Cohomology ring structure Other homology and cohomology theories (SHT, K-theory)
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Develop Tools
We want to compute these algebraic structures Develop theoretical tools Postnikov towers Spectral sequences Stable homotopy theory “ad-hoc”
Algorithmic/mechanical approach
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Develop Tools
We want to compute these algebraic structures Develop theoretical tools Postnikov towers Spectral sequences Stable homotopy theory “ad-hoc”
Algorithmic/mechanical approach
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Develop Tools
We want to compute these algebraic structures Develop theoretical tools Postnikov towers Spectral sequences Stable homotopy theory “ad-hoc”
Algorithmic/mechanical approach
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Develop Tools
We want to compute these algebraic structures Develop theoretical tools Postnikov towers Spectral sequences Stable homotopy theory “ad-hoc”
Algorithmic/mechanical approach
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Develop Tools
We want to compute these algebraic structures Develop theoretical tools Postnikov towers Spectral sequences Stable homotopy theory “ad-hoc”
Algorithmic/mechanical approach
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Develop Tools
We want to compute these algebraic structures Develop theoretical tools Postnikov towers Spectral sequences Stable homotopy theory “ad-hoc”
Algorithmic/mechanical approach
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Develop Tools
We want to compute these algebraic structures Develop theoretical tools Postnikov towers Spectral sequences Stable homotopy theory “ad-hoc”
Algorithmic/mechanical approach
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Recall the Usual Procedure
1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R).
3
Turn into a chain complex using the boundary maps ∂.
4
Take homology of the complex.
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Recall the Usual Procedure
1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R).
3
Turn into a chain complex using the boundary maps ∂.
4
Take homology of the complex.
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Recall the Usual Procedure
1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R).
3
Turn into a chain complex using the boundary maps ∂.
4
Take homology of the complex.
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Recall the Usual Procedure
1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R).
3
Turn into a chain complex using the boundary maps ∂.
4
Take homology of the complex.
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Recall the Usual Procedure
1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R).
3
Turn into a chain complex using the boundary maps ∂.
4
Take homology of the complex.
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Cell Complex v1
v2
v0
v3
Start with 0-cells = points
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Cell Complex v1 b a
v2 c
v0
e
d v3
Attach 1-cells = edges
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Cell Complex
Attach 2-cells = faces Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Cell Complex
And continue...
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Translate Into Algebra Next, we take all our cells and group them together by dimension Cn (X , R) = R[n-dimensional cells]. In our example
C0 (X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3 C1 (X ) = Ra ⊕ Rb ⊕ Rc ⊕ Rd ⊕ Re C2 (X ) = RF ⊕ RG Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Translate Into Algebra Next, we take all our cells and group them together by dimension Cn (X , R) = R[n-dimensional cells]. In our example
C0 (X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3 C1 (X ) = Ra ⊕ Rb ⊕ Rc ⊕ Rd ⊕ Re C2 (X ) = RF ⊕ RG Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Translate Into Algebra Next, we take all our cells and group them together by dimension Cn (X , R) = R[n-dimensional cells]. In our example
C0 (X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3 C1 (X ) = Ra ⊕ Rb ⊕ Rc ⊕ Rd ⊕ Re C2 (X ) = RF ⊕ RG Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Translate Into Algebra Next, we take all our cells and group them together by dimension Cn (X , R) = R[n-dimensional cells]. In our example
C0 (X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3 C1 (X ) = Ra ⊕ Rb ⊕ Rc ⊕ Rd ⊕ Re C2 (X ) = RF ⊕ RG Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Translate Into Algebra Next, we take all our cells and group them together by dimension Cn (X , R) = R[n-dimensional cells]. In our example
C0 (X ) = Rv0 ⊕ Rv1 ⊕ Rv2 ⊕ Rv3 C1 (X ) = Ra ⊕ Rb ⊕ Rc ⊕ Rd ⊕ Re C2 (X ) = RF ⊕ RG Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Boundary Maps ∂
We order the vertices Get boundary maps ∂ : Cn (X ) → Cn−1 (X ) In general ∂[v0 , . . . , vn ] =
n X
(−1)i [v0 , . . . , vˆi , . . . , vn ]
i=0
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Boundary Maps ∂
We order the vertices Get boundary maps ∂ : Cn (X ) → Cn−1 (X ) In general ∂[v0 , . . . , vn ] =
n X
(−1)i [v0 , . . . , vˆi , . . . , vn ]
i=0
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Boundary Maps ∂
We order the vertices Get boundary maps ∂ : Cn (X ) → Cn−1 (X ) In general ∂[v0 , . . . , vn ] =
n X
(−1)i [v0 , . . . , vˆi , . . . , vn ]
i=0
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Boundary Maps ∂ In our example Order is v0 < v1 < v2 < v3 ∂a = v1 − v0 ∂F = b − c + a
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Boundary Maps ∂ In our example Order is v0 < v1 < v2 < v3 ∂a = v1 − v0 ∂F = b − c + a
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Boundary Maps ∂ In our example Order is v0 < v1 < v2 < v3 ∂a = v1 − v0 ∂F = b − c + a
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Boundary Maps ∂ In our example Order is v0 < v1 < v2 < v3 ∂a = v1 − v0 ∂F = b − c + a
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Homology Groups Prove ∂ 2 = 0 Gives a chain complex ∂
∂
· · · → Cn+1 (X ) − → Cn (X ) − → Cn−1 (X ) → · · · Homology of the complex is simplicial homology ker ∂ : Cn (X ) → Cn−1 (X ) Hn (X , R) = im ∂ : Cn+1 (X ) → Cn (X )
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Homology Groups Prove ∂ 2 = 0 Gives a chain complex ∂
∂
· · · → Cn+1 (X ) − → Cn (X ) − → Cn−1 (X ) → · · · Homology of the complex is simplicial homology ker ∂ : Cn (X ) → Cn−1 (X ) Hn (X , R) = im ∂ : Cn+1 (X ) → Cn (X )
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Homology Groups Prove ∂ 2 = 0 Gives a chain complex ∂
∂
· · · → Cn+1 (X ) − → Cn (X ) − → Cn−1 (X ) → · · · Homology of the complex is simplicial homology ker ∂ : Cn (X ) → Cn−1 (X ) Hn (X , R) = im ∂ : Cn+1 (X ) → Cn (X )
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Computation
Question Can we program a machine to do all of this for us? 1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R).
3
Turn into a chain complex using the boundary maps ∂.
4
Take homology of the complex.
5
As long as R is a field
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Computation
Question Can we program a machine to do all of this for us? 1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R).
3
Turn into a chain complex using the boundary maps ∂.
4
Take homology of the complex.
5
As long as R is a field
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Computation
Question Can we program a machine to do all of this for us? 1
Create a cell complex X .
2
Create a graded R-module Cn (X ; R). Yes
3
Turn into a chain complex using the boundary maps ∂. Yes
4
Take homology of the complex. Yes
5
As long as R is a field
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Philosophy of Algebraic Topology Simplicial Homology
Computation
Question Can we program a machine to do all of this for us? 1
Create a cell complex X . Yes!
2
Create a graded R-module Cn (X ; R). Yes
3
Turn into a chain complex using the boundary maps ∂. Yes
4
Take homology of the complex. Yes
5
As long as R is a field
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Given a space X , how can we turn over the computation of homology to a machine? Usually have some idea of distance on X , and we use it to cover X .
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Given a space X , how can we turn over the computation of homology to a machine? Usually have some idea of distance on X , and we use it to cover X .
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Useful Result
Notation U = {Ua }a∈A is an open cover of X ˘ C(U) is the Cech complex of U Vertex set A a0 , . . . , an span an n-simplex if Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Useful Result
Notation U = {Ua }a∈A is an open cover of X ˘ C(U) is the Cech complex of U Vertex set A a0 , . . . , an span an n-simplex if Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Useful Result
Notation U = {Ua }a∈A is an open cover of X ˘ C(U) is the Cech complex of U Vertex set A a0 , . . . , an span an n-simplex if Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Useful Result
Notation U = {Ua }a∈A is an open cover of X ˘ C(U) is the Cech complex of U Vertex set A a0 , . . . , an span an n-simplex if Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Useful Result
Notation U = {Ua }a∈A is an open cover of X ˘ C(U) is the Cech complex of U Vertex set A a0 , . . . , an span an n-simplex if Ua0 ∩ · · · ∩ Uan 6= ∅
Robert Hank
Computational Algebraic Topology
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Review Basic Definitions Example
Useful Result Theorem Suppose U is finite Arbitrary intersections n \
Uai
i=1
are either contractible or empty.
Robert Hank
Computational Algebraic Topology
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Review Basic Definitions Example
Useful Result Theorem Suppose U is finite Arbitrary intersections n \
Uai
i=1
are either contractible or empty.
Robert Hank
Computational Algebraic Topology
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Review Basic Definitions Example
Useful Result Theorem Suppose U is finite Arbitrary intersections n \
Uai
i=1
are either contractible or empty.
Robert Hank
Computational Algebraic Topology
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Review Basic Definitions Example
Useful Result Theorem Suppose U is finite Arbitrary intersections n \
Uai
i=1
are either contractible or empty. Then C(U) is homotopy equivalent to X .
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
What is a Point Cloud? Finite collection of points
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
What is a Point Cloud? Finite collection of points
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
What is a Point Cloud? Finite collection of points
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
What is a Point Cloud? Finite collection of points
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Sampling
Given a space X , take a “random sampling”
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Sampling Given a space X , take a “random sampling”
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Sampling Given a space X , take a “random sampling”
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Sampling We would like to recover X from the sample
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Data
Data samples can also create point clouds We would like to understand the shape the data takes Insights into the shape can be very useful for interpreting the data
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Data
Data samples can also create point clouds We would like to understand the shape the data takes Insights into the shape can be very useful for interpreting the data
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Data
Data samples can also create point clouds We would like to understand the shape the data takes Insights into the shape can be very useful for interpreting the data
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Metric
In practice, our space X will come equipped with a metric d We can use the metric to construct a complex This can be programmed so a computer can run the algorithm
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Metric
In practice, our space X will come equipped with a metric d We can use the metric to construct a complex This can be programmed so a computer can run the algorithm
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Metric
In practice, our space X will come equipped with a metric d We can use the metric to construct a complex This can be programmed so a computer can run the algorithm
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Given a point cloud X , VR(X , ) is the Vietoris-Rips complex associated to the parameter . Vertex set is X x0 , . . . , xn span an n-simplex if the distance between any pair is ≤ .
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Given a point cloud X , VR(X , ) is the Vietoris-Rips complex associated to the parameter . Vertex set is X x0 , . . . , xn span an n-simplex if the distance between any pair is ≤ .
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Given a point cloud X , VR(X , ) is the Vietoris-Rips complex associated to the parameter . Vertex set is X x0 , . . . , xn span an n-simplex if the distance between any pair is ≤ .
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Problem: If we start with X as the entire space, this complex is really large
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Vietoris-Rips
Problem: If we start with X as the entire space, this complex is really large
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx +
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx +
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx +
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx +
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx + l0 , . . . , ln span an n-simplex if d(x, li ) ≤ mx + for every i This can be automated and gives smaller complexes Efficiency adjustments give other types of witness complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx + l0 , . . . , ln span an n-simplex if d(x, li ) ≤ mx + for every i This can be automated and gives smaller complexes Efficiency adjustments give other types of witness complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx + l0 , . . . , ln span an n-simplex if d(x, li ) ≤ mx + for every i This can be automated and gives smaller complexes Efficiency adjustments give other types of witness complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Witness Complexes
Solution: Sample X with a set L of “landmark points” Given x ∈ X , let mx be the distance from x to L Choose x is a “witness” to li if d(x, li ) ≤ mx + l0 , . . . , ln span an n-simplex if d(x, li ) ≤ mx + for every i This can be automated and gives smaller complexes Efficiency adjustments give other types of witness complexes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Outline 1
Review Philosophy of Algebraic Topology Simplicial Homology
2
Basic Definitions ˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
3
Example
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
˘ In all our complexes (Cech, Vietoris-Rips, witness), we have a changing parameter We get different complexes C• (X , ) depending on But if < 0 we get an inclusion C• (X , ) → C• (X , 0 ) Eventually, is so large that C• (X , ) doesn’t change
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
˘ In all our complexes (Cech, Vietoris-Rips, witness), we have a changing parameter We get different complexes C• (X , ) depending on But if < 0 we get an inclusion C• (X , ) → C• (X , 0 ) Eventually, is so large that C• (X , ) doesn’t change
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
˘ In all our complexes (Cech, Vietoris-Rips, witness), we have a changing parameter We get different complexes C• (X , ) depending on But if < 0 we get an inclusion C• (X , ) → C• (X , 0 ) Eventually, is so large that C• (X , ) doesn’t change
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Idea
˘ In all our complexes (Cech, Vietoris-Rips, witness), we have a changing parameter We get different complexes C• (X , ) depending on But if < 0 we get an inclusion C• (X , ) → C• (X , 0 ) Eventually, is so large that C• (X , ) doesn’t change
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Barcodes
If our base ring is a field F Our complexes are vector spaces We have algebraic algorithms to decompose vector spaces Make a countable, order-preserving choice of paramaters , f : N → R Represent the change in vector spaces via barcodes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Barcodes
If our base ring is a field F Our complexes are vector spaces We have algebraic algorithms to decompose vector spaces Make a countable, order-preserving choice of paramaters , f : N → R Represent the change in vector spaces via barcodes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Barcodes
If our base ring is a field F Our complexes are vector spaces We have algebraic algorithms to decompose vector spaces Make a countable, order-preserving choice of paramaters , f : N → R Represent the change in vector spaces via barcodes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Barcodes
If our base ring is a field F Our complexes are vector spaces We have algebraic algorithms to decompose vector spaces Make a countable, order-preserving choice of paramaters , f : N → R Represent the change in vector spaces via barcodes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Barcodes
If our base ring is a field F Our complexes are vector spaces We have algebraic algorithms to decompose vector spaces Make a countable, order-preserving choice of paramaters , f : N → R Represent the change in vector spaces via barcodes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Barcodes
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Problems
How do we handle multiple parameters changing (e.g. L)? Multidimensional Persistence Persistence relies on having nested maps C• (X , n) → C• (X , n + 1) What if we can’t get these? Zigzag Persistence
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Problems
How do we handle multiple parameters changing (e.g. L)? Multidimensional Persistence Persistence relies on having nested maps C• (X , n) → C• (X , n + 1) What if we can’t get these? Zigzag Persistence
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Problems
How do we handle multiple parameters changing (e.g. L)? Multidimensional Persistence Persistence relies on having nested maps C• (X , n) → C• (X , n + 1) What if we can’t get these? Zigzag Persistence
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Problems
How do we handle multiple parameters changing (e.g. L)? Multidimensional Persistence Persistence relies on having nested maps C• (X , n) → C• (X , n + 1) What if we can’t get these? Zigzag Persistence
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
Problems
How do we handle multiple parameters changing (e.g. L)? Multidimensional Persistence Persistence relies on having nested maps C• (X , n) → C• (X , n + 1) What if we can’t get these? Zigzag Persistence
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
˘ Cech complex Point Clouds Complexes of Point Clouds Persistence
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Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Analyze photographs Lee, Pedersen, and Mumford in 2003 Thank you to Gunnar Carlsson for the images!
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Analyze photographs Lee, Pedersen, and Mumford in 2003 Thank you to Gunnar Carlsson for the images!
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Analyze photographs Lee, Pedersen, and Mumford in 2003 Thank you to Gunnar Carlsson for the images!
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
Example: Image Statistics
Robert Hank
Computational Algebraic Topology
Review Basic Definitions Example
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Robert Hank
Computational Algebraic Topology
