5. Persistence Diagrams...5. Persistence Diagrams Vin de Silva Pomona College CBMS Lecture Series,...
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5. Persistence Diagrams
Vin de SilvaPomona College
CBMS Lecture Series, Macalester College, June 2017
Vin de Silva Pomona College 5. Persistence Diagrams
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Persistence modules
Persistence modules
A diagram of vector spaces and linear maps
V0// V1
// . . . // Vn
is called a persistence module.
Vin de Silva Pomona College 5. Persistence Diagrams
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Persistence modules
Persistence modules
A diagram of vector spaces and linear maps
V0// V1
// . . . // Vn
is called a persistence module.
Example
Let
S0⊆// S1
⊆// . . .
⊆// Sn
be a nested sequence of simplicial complexes. Then
Hk(S0) // Hk(S1) // . . . // Hk(Sn)
is a persistence module. The arrows denote the maps in homology induced bythe inclusions Sk−1 ⊆ Sk .
Vin de Silva Pomona College 5. Persistence Diagrams
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Persistence modules
Persistence modules
A diagram of vector spaces and linear maps
V0// V1
// . . . // Vn
is called a persistence module.
Example
Let X be a finite metric space and let
r0 < r1 < · · · < rn
be the set of values taken by the metric. Then
Hk(VR(X , r0)) // Hk(VR(X , r1)) // . . . // Hk(VR(X , rn))
is a persistence module. The arrows denote the maps in homology induced bythe inclusions VR(X , rk−1) ⊆ VR(X , rk).
Vin de Silva Pomona College 5. Persistence Diagrams
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Persistence modules
Persistence modules
A diagram of vector spaces and linear maps
V0// V1
// . . . // Vn
is called a persistence module.
Example
Let M be a compact smooth manifold and let f : M → R be a smooth functionwith finitely many critical values
t0 < t1 < · · · < tn.
Let M t = f −1(−∞, t]. Then
Hk(M t0 ) // Hk(M t1 ) // . . . // Hk(M tn )
is a persistence module. The arrows denote the maps in homology induced bythe inclusions M tk−1 ⊆ M tk .
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Persistence modules
Persistence modules
A persistence module indexed by N is an infinite diagram
V0// V1
// . . . // Vn// . . .
of vector spaces and linear maps.
Zomorodian, Carlsson (2002)
This is the same thing as a graded module over the polynomial ring F[t].
Field elements act on each Vk by scalar multiplication. The ring element t actsby applying the map Vk → Vk+1.
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The rank invariant
n=0
Diagrams of the formV0
are classified up to isomorphism by a single numerical invariant:
dim(V0)
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The rank invariant
n=0
Diagrams of the formV0
are classified up to isomorphism by a single numerical invariant:
r(0, 0) = rank(V0 → V0) = dim(V0)
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The rank invariant
n=1
Diagrams of the form
V0// V1
are classified up to isomorphism by three numerical invariants:
dim(V0)
rank(V0 → V1)
dim(V1)
Proof
Represent the map by a matrix. Changes of basis of V0,V1 correspond to columnand row operations. Every matrix can be reduced to the block form:[
Ir 00 0
]These block forms are precisely classified by the three invariants.
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The rank invariant
n=1
Diagrams of the form
V0// V1
are classified up to isomorphism by three numerical invariants:
r(0, 0) = dim(V0) = rank(V0 → V0)
r(0, 1) = rank(V0 → V1)
r(1, 1) = dim(V1) = rank(V1 → V1)
Proof
Represent the map by a matrix. Changes of basis of V0,V1 correspond to columnand row operations. Every matrix can be reduced to the block form:[
Ir 00 0
]These block forms are precisely classified by the three invariants.
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The rank invariant
Theorem (Zomorodian, Carlsson, 2002)
Diagrams of the form
V0// V1
// . . . // Vn
are classified up to isomorphism by the invariants
r(i , j) = rank(Vi → Vj)
for all 0 ≤ i ≤ j ≤ n.
Plot these as points-with-multiplicity on a ‘rank diagram’:
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 r(i, j)
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The rank invariant
Theorem (Zomorodian, Carlsson, 2002)
Diagrams of the form
V0// V1
// . . . // Vn
are classified up to isomorphism by the invariants
r(i , j) = rank(Vi → Vj)
for all 0 ≤ i ≤ j ≤ n.
Plot these as points-with-multiplicity on a ‘rank diagram’:
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 r(i, j)
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Interval decomposition
Direct sums
Given two persistence modules
V0// V1
// . . . // Vn
W0// W1
// . . . // Wn
we can form their direct sum:
V0 ⊕W0// V1 ⊕W1
// . . . // Vn ⊕Wn
Direct sum decomposition
Given a persistence module
V0// V1
// . . . // Vn
can it be expressed as the direct sum of two non-trivial persistence modules? Ifit can’t, then it is indecomposable.
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Interval decomposition
Interval modules
The interval module I (i , j) is defined to be the persistence module with
Vk =
{F if i ≤ k ≤ j
0 otherwise
and all maps F→ F being the identity (the other maps necessarily being zero).
Here are examples of interval modules with n = 5:
0 // 0 // F 1 // F 1 // F // 0
0 // 0 // F 1 // F 1 // F // F
F 1 // F 1 // F 1 // F 1 // F // F
Fact
Interval modules are indecomposable.
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Interval decomposition
Interval modules
The interval module I (i , j) is defined to be the persistence module with
Vk =
{F if i ≤ k ≤ j
0 otherwise
and all maps F→ F being the identity (the other maps necessarily being zero).
Here are examples of interval modules with n = 5:
0 // 0 // F 1 // F 1 // F // 0
0 // 0 // F 1 // F 1 // F // F
F 1 // F 1 // F 1 // F 1 // F // F
Fact
Interval modules are indecomposable.
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Interval decomposition
Theorem (Gabriel 1970; Zomorodian & Carlsson 2002; etc.)
Every (finite-dimensional) persistence module is isomorphic to a (finite) directsum of interval modules. The multiplicity
m(i , j)
of the summand I (i , j) is an invariant of the persistence module.
Proof
Zomorodian & Carlsson use the classification theorem for finitely-generatedgraded modules over F[t].
Plot the multiplicities as points-with-multiplicity on a ‘persistence diagram’:
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 m(i, j)
Vin de Silva Pomona College 5. Persistence Diagrams
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Interval decomposition
Theorem (Gabriel 1970; Zomorodian & Carlsson 2002; etc.)
Every (finite-dimensional) persistence module is isomorphic to a (finite) directsum of interval modules. The multiplicity
m(i , j)
of the summand I (i , j) is an invariant of the persistence module.
Proof
Zomorodian & Carlsson use the classification theorem for finitely-generatedgraded modules over F[t].
Plot the multiplicities as points-with-multiplicity on a ‘persistence diagram’:
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 m(i, j)
Vin de Silva Pomona College 5. Persistence Diagrams
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Interval decomposition
Theorem (Gabriel 1970; Zomorodian & Carlsson 2002; etc.)
Every (finite-dimensional) persistence module is isomorphic to a (finite) directsum of interval modules. The multiplicity
m(i , j)
of the summand I (i , j) is an invariant of the persistence module.
Proof
Zomorodian & Carlsson use the classification theorem for finitely-generatedgraded modules over F[t].
Plot the multiplicities as points-with-multiplicity on a ‘persistence diagram’:
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 m(i, j)
Vin de Silva Pomona College 5. Persistence Diagrams
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Relationship between the two diagrams
Inverse transformations
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 r(i, j)
//oo
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 m(i, j)
rank diagram ← persistence diagram
Each summand I (p, q) contributes to r(i , j) whenever p ≤ i ≤ j ≤ q. Thus:
r(i , j) =∑i
p=0
∑nq=j m(p, q)
rank diagram → persistence diagram
The inverse transformation is
m(i , j) = r(i , j)− r(i − 1, j)− r(i , j + 1) + r(i − 1, j + 1)
by the inclusion exclusion principle.
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Relationship between the two diagrams
Inverse transformations
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 r(i, j)
//oo
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 m(i, j)
rank diagram ← persistence diagram
Each summand I (p, q) contributes to r(i , j) whenever p ≤ i ≤ j ≤ q. Thus:
r(i , j) =∑i
p=0
∑nq=j m(p, q)
rank diagram → persistence diagram
The inverse transformation is
m(i , j) = r(i , j)− r(i − 1, j)− r(i , j + 1) + r(i − 1, j + 1)
by the inclusion exclusion principle.
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Relationship between the two diagrams
Inverse transformations
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 r(i, j)
//oo
• • • • • 4
• • • • 3
• • • 2
• • 1
• 0
0 1 2 3 4 m(i, j)
rank diagram ← persistence diagram
Each summand I (p, q) contributes to r(i , j) whenever p ≤ i ≤ j ≤ q. Thus:
r(i , j) =∑i
p=0
∑nq=j m(p, q)
rank diagram → persistence diagram
The inverse transformation is
m(i , j) = r(i , j)− r(i − 1, j)− r(i , j + 1) + r(i − 1, j + 1)
by the inclusion exclusion principle.
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Relationship between the two diagrams
Inverse transformations
0 0 0 0 0 4
0 2 2 2 3
1 3 3 2
1 3 1
1 0
0 1 2 3 4 r(i, j)
//oo
◦ ◦ ◦ ◦ ◦ 4
◦ 2 ◦ ◦ 3
1 ◦ ◦ 2
◦ ◦ 1
◦ 0
0 1 2 3 4 m(i, j)
rank diagram ← persistence diagram
Each summand I (p, q) contributes to r(i , j) whenever p ≤ i ≤ j ≤ q. Thus:
r(i , j) =∑i
p=0
∑nq=j m(p, q)
rank diagram → persistence diagram
The inverse transformation is
m(i , j) = r(i , j)− r(i − 1, j)− r(i , j + 1) + r(i − 1, j + 1)
by the inclusion exclusion principle.
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Plotting the persistence diagram
Plotting conventions
For persistence modules such as the Vietoris–Rips homology of a finite metricspace, constructed with respect to a set of critical values
r0 < r1 < · · · < rn
we adopt the following convention. Each summand I (i , j) is drawn as
• a half-open interval [ri , rj+1) in the barcode; or
• a point (ri , rj+1) in the persistence diagram.
We set rn+1 = +∞ by convention.Persistence diagram
smallscale
largescale
Barcode0 0.5 10
0.5
1
small scale
large scale
Persistence diagramintervals [b,d)points (b,d)
Persistence diagram
smallscale
largescale
Barcode0 0.5 10
0.5
1
small scale
large scale
Persistence diagramintervals [b,d)points (b,d)Vin de Silva Pomona College 5. Persistence Diagrams
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Vin de Silva Pomona College 5. Persistence Diagrams
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The ELZ Persistence Algorithm
Edelsbrunner, Letscher, Zomorodian (2000)
Given a nested sequence of simplicial complexes
S0⊆// S1
⊆// . . .
⊆// Sn
we wish to compute the persistence diagram of
Hd(S0) // Hd(S1) // . . . // Hd(Sn).
We compute it for all d simultaneously (up to some maximum dimension).
Input
A sequence of ‘cells’ σ0, σ1, . . . , σn. For each cell we are given:
• the dimension d = dim(σk);
• the boundary ∂[σk ], as a cycle built from existing (d − 1)-cells.
Output
The persistence diagram for this nested sequence of formal cell-complexes.
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The ELZ Persistence Algorithm
Edelsbrunner, Letscher, Zomorodian (2000)
Given a nested sequence of simplicial complexes
S0⊆// S1
⊆// . . .
⊆// Sn
we wish to compute the persistence diagram of
Hd(S0) // Hd(S1) // . . . // Hd(Sn).
We compute it for all d simultaneously (up to some maximum dimension).
Input
A sequence of ‘cells’ σ0, σ1, . . . , σn. For each cell we are given:
• the dimension d = dim(σk);
• the boundary ∂[σk ], as a cycle built from existing (d − 1)-cells.
Output
The persistence diagram for this nested sequence of formal cell-complexes.
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The ELZ Persistence Algorithm
Data structure
We maintain an upper-triangular square matrix, size equal to the number of cellsprocessed so far. Columns are coloured green, yellow, red.
colour content leading coefficient
green d-cycle αk 1yellow d-cycle βk 1red d-chain γk non-zero
Here d is the dimension of the cell σk whose column it is.
We maintain a bijection between the yellow and red columns.If yellow column k is paired with red column `, then k < ` and
∂γ` = βk .
Readout
Each green column j corresponds to an interval [j ,+∞).Each yellow–red pair (k, `) corresponds to an interval [k, `).
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The ELZ Persistence Algorithm
Matrix
1 2 3 12 23 13
1 1 -1
2 1
3 1
12 ∗ 1
23
13
1
23
Chains
green : α0 = [1], α2 = [3]
yellow : β1 = [2]− [1]
red : γ3 = [1, 2]
Persistence intervals
[0,+∞), [1, 3), [2,+∞)
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The ELZ Persistence Algorithm
Lemma
The green and yellow columns form a basis for the space of all cycles.The yellow columns form a basis for the space of all boundaries.
Cycle reduction
Using the table, any cycle ψ can be uniquely written
ψ = ∂ρ or ψ = ∂ρ+ φ
ρ is a combination of red columns, φ is a cycle whose leading simplex is green.
Iterative method
Initialise ρ = 0.Consider ψ − ∂ρ.
• If it is zero then STOP.
• If its leading simplex is green, then set φ = ψ − ∂ρ and STOP.
• If its leading simplex is yellow, then add to ρ the appropriate multiple ofthe associated red column to eliminate that leading term.
Repeat.
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The ELZ Persistence Algorithm
Lemma
The green and yellow columns form a basis for the space of all cycles.The yellow columns form a basis for the space of all boundaries.
Cycle reduction
Using the table, any cycle ψ can be uniquely written
ψ = ∂ρ or ψ = ∂ρ+ φ
ρ is a combination of red columns, φ is a cycle whose leading simplex is green.
Iterative method
Initialise ρ = 0.Consider ψ − ∂ρ.
• If it is zero then STOP.
• If its leading simplex is green, then set φ = ψ − ∂ρ and STOP.
• If its leading simplex is yellow, then add to ρ the appropriate multiple ofthe associated red column to eliminate that leading term.
Repeat.
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The ELZ Persistence Algorithm
Lemma
The green and yellow columns form a basis for the space of all cycles.The yellow columns form a basis for the space of all boundaries.
Cycle reduction
Using the table, any cycle ψ can be uniquely written
ψ = ∂ρ or ψ = ∂ρ+ φ
ρ is a combination of red columns, φ is a cycle whose leading simplex is green.
Iterative method
Initialise ρ = 0.Consider ψ − ∂ρ.
• If it is zero then STOP.
• If its leading simplex is green, then set φ = ψ − ∂ρ and STOP.
• If its leading simplex is yellow, then add to ρ the appropriate multiple ofthe associated red column to eliminate that leading term.
Repeat.
Vin de Silva Pomona College 5. Persistence Diagrams
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The ELZ Persistence Algorithm
Update Step
Consider the next simplex σk .
If ∂[σk ] was already a boundary: The cycle reduction lemma gives
∂[σk ] = ∂ρ.
We get a new green cycleαk = [σk ]− ρ.
If ∂[σk ] was not previously a boundary: The cycle reduction lemma gives
∂[σk ] = ∂ρ+ φ
where the leading term of φ is c[σj ] for some scalar c.Change the j-th column to yellow, and colour the k-th column red. Set
βj = c−1φ, γk = c−1([σk ]− ρ)
(discarding αj). Pair the columns j and k.
Vin de Silva Pomona College 5. Persistence Diagrams
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The ELZ Persistence Algorithm
Update Step
Consider the next simplex σk .
If ∂[σk ] was already a boundary: The cycle reduction lemma gives
∂[σk ] = ∂ρ.
We get a new green cycleαk = [σk ]− ρ.
If ∂[σk ] was not previously a boundary: The cycle reduction lemma gives
∂[σk ] = ∂ρ+ φ
where the leading term of φ is c[σj ] for some scalar c.Change the j-th column to yellow, and colour the k-th column red. Set
βj = c−1φ, γk = c−1([σk ]− ρ)
(discarding αj). Pair the columns j and k.
Vin de Silva Pomona College 5. Persistence Diagrams
![Page 34: 5. Persistence Diagrams...5. Persistence Diagrams Vin de Silva Pomona College CBMS Lecture Series, Macalester College, June 2017 Vin de Silva Pomona College 5. Persistence Diagrams](https://reader034.fdocuments.in/reader034/viewer/2022051808/600ea8901589224d2f1335fc/html5/thumbnails/34.jpg)
The ELZ Persistence Algorithm
Update Step
Consider the next simplex σk .
If ∂[σk ] was already a boundary: The cycle reduction lemma gives
∂[σk ] = ∂ρ.
We get a new green cycleαk = [σk ]− ρ.
If ∂[σk ] was not previously a boundary: The cycle reduction lemma gives
∂[σk ] = ∂ρ+ φ
where the leading term of φ is c[σj ] for some scalar c.Change the j-th column to yellow, and colour the k-th column red. Set
βj = c−1φ, γk = c−1([σk ]− ρ)
(discarding αj). Pair the columns j and k.
Vin de Silva Pomona College 5. Persistence Diagrams
![Page 35: 5. Persistence Diagrams...5. Persistence Diagrams Vin de Silva Pomona College CBMS Lecture Series, Macalester College, June 2017 Vin de Silva Pomona College 5. Persistence Diagrams](https://reader034.fdocuments.in/reader034/viewer/2022051808/600ea8901589224d2f1335fc/html5/thumbnails/35.jpg)
The ELZ Persistence Algorithm
Matrix
1 2 3 12 23 13
1 1 -1
2 1
3 1
12 ∗ 1
23
13
1
23
Chains
green : α0 = [1], α2 = [3]
yellow : β1 = [2]− [1]
red : γ3 = [1, 2]
Persistence intervals
[0,+∞), [1, 3), [2,+∞)
Vin de Silva Pomona College 5. Persistence Diagrams
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The ELZ Persistence Algorithm
Matrix
1 2 3 12 23 13
1 1 -1
2 1 -1
3 1
12 ∗ 1
23 ∗ 1
13
1
23
Chains
green : α0 = [1]
yellow : β1 = [2]− [1], β2 = [3]− [2]
red : γ3 = [1, 2], γ4 = [2, 3]
Persistence intervals
[0,+∞), [1, 3), [2, 4)
Vin de Silva Pomona College 5. Persistence Diagrams
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The ELZ Persistence Algorithm
Matrix
1 2 3 12 23 13
1 1 -1
2 1 -1
3 1
12 ∗ 1 -1
23 ∗ 1 -1
13 1
1
23
Chains
green : α0 = [1], α5 = [1, 3]− [2, 3]− [1, 2]
yellow : β1 = [2]− [1], β2 = [3]− [2]
red : γ3 = [1, 2], γ4 = [2, 3]
Persistence intervals
[0,+∞), [1, 3), [2, 4), [5,+∞)
Vin de Silva Pomona College 5. Persistence Diagrams
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Vin de Silva Pomona College 5. Persistence Diagrams