5. persistence diagrams...5. persistence diagrams vin de silva pomona college cbms lecture series,...

5. Persistence Diagrams

Vin de SilvaPomona College

CBMS Lecture Series, Macalester College, June 2017

Vin de Silva Pomona College 5. Persistence Diagrams

Persistence modules

Persistence modules

A diagram of vector spaces and linear maps

V0// V1

// . . . // Vn

is called a persistence module.


Persistence modules

Persistence modules


V0// V1

// . . . // Vn


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 .


Persistence modules

Persistence modules


V0// V1

// . . . // Vn


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).


Persistence modules

Persistence modules


V0// V1

// . . . // Vn


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 .


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.


The rank invariant

n=0

Diagrams of the formV0

are classified up to isomorphism by a single numerical invariant:

dim(V0)


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)


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.


The rank invariant

n=1


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.


The rank invariant

Theorem (Zomorodian, Carlsson, 2002)


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)


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.



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.



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)


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.


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.


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

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.



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, `).



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,+∞)



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.



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.



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]


[0,+∞), [1, 3), [2,+∞)



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]


[0,+∞), [1, 3), [2, 4)



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]


[0,+∞), [1, 3), [2, 4), [5,+∞)


5. persistence diagrams...5. persistence diagrams vin de silva pomona college cbms lecture series,...

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