5. Persistence Diagrams

Transcription

1 Vin de Silva Pomona College CBMS Lecture Series, Macalester College, June 207

2 Persistence modules Persistence modules A diagram of vector spaces and linear maps V 0 V... V n is called a persistence module.

3 Persistence modules Persistence modules A diagram of vector spaces and linear maps V 0 V... V n is called a persistence module. Example Let S 0 S... be a nested sequence of simplicial complexes. Then S n H k (S 0) H k (S )... H k (S n) is a persistence module. The arrows denote the maps in homology induced by the inclusions S k S k.

4 Persistence modules Persistence modules A diagram of vector spaces and linear maps is called a persistence module. Example Let X be a finite metric space and let V 0 V... V n r 0 < r < < r n be the set of values taken by the metric. Then H k (VR(X, r 0)) H k (VR(X, r ))... H k (VR(X, r n)) is a persistence module. The arrows denote the maps in homology induced by the inclusions VR(X, r k ) VR(X, r k ).

5 Persistence modules Persistence modules A diagram of vector spaces and linear maps is called a persistence module. Example V 0 V... V n Let M be a compact smooth manifold and let f : M R be a smooth function with finitely many critical values Let M t = f (, t]. Then t 0 < t < < t n. H k (M t 0 ) H k (M t )... H k (M tn ) is a persistence module. The arrows denote the maps in homology induced by the inclusions M t k M t k.

6 Persistence modules Persistence modules A persistence module indexed by N is an infinite diagram of vector spaces and linear maps. Zomorodian, Carlsson (2002) V 0 V... V n... This is the same thing as a graded module over the polynomial ring F[t]. Field elements act on each V k by scalar multiplication. The ring element t acts by applying the map V k V k+.

7 The rank invariant n=0 Diagrams of the form are classified up to isomorphism by a single numerical invariant: dim(v 0) V 0

8 The rank invariant n=0 Diagrams of the form are classified up to isomorphism by a single numerical invariant: r(0, 0) = rank(v 0 V 0) = dim(v 0) V 0

9 The rank invariant n= Diagrams of the form V 0 V are classified up to isomorphism by three numerical invariants: dim(v 0) rank(v 0 V ) dim(v ) Proof Represent the map by a matrix. Changes of basis of V 0, V correspond to column and row operations. Every matrix can be reduced to the block form: [ Ir 0 ] 0 0 These block forms are precisely classified by the three invariants.

10 The rank invariant n= Diagrams of the form V 0 V are classified up to isomorphism by three numerical invariants: r(0, 0) = dim(v 0) = rank(v 0 V 0) r(0, ) = rank(v 0 V ) r(, ) = dim(v ) = rank(v V ) Proof Represent the map by a matrix. Changes of basis of V 0, V correspond to column and row operations. Every matrix can be reduced to the block form: [ Ir 0 ] 0 0 These block forms are precisely classified by the three invariants.

11 The rank invariant Theorem (Zomorodian, Carlsson, 2002) Diagrams of the form V 0 V... V n are classified up to isomorphism by the invariants r(i, j) = rank(v i V j ) for all 0 i j n.

12 The rank invariant Theorem (Zomorodian, Carlsson, 2002) Diagrams of the form V 0 V... V n are classified up to isomorphism by the invariants for all 0 i j n. r(i, j) = rank(v i V j ) Plot these as points-with-multiplicity on a rank diagram : r(i, j)

13 Interval decomposition Direct sums Given two persistence modules V 0 V... V n we can form their direct sum: Direct sum decomposition W 0 W... W n V 0 W 0 V W... V n W n Given a persistence module V 0 V... V n can it be expressed as the direct sum of two non-trivial persistence modules? If it can t, then it is indecomposable.

14 Interval decomposition Interval modules The interval module I (i, j) is defined to be the persistence module with { F if i k j V k = 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 0 0 F F F F 0 F F F F F F F F

15 Interval decomposition Interval modules The interval module I (i, j) is defined to be the persistence module with { F if i k j V k = 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 0 0 F F F F 0 F F F F F F F F Fact Interval modules are indecomposable.

16 Interval decomposition Theorem (Gabriel 970; Zomorodian & Carlsson 2002; etc.) Every (finite-dimensional) persistence module is isomorphic to a (finite) direct sum of interval modules. The multiplicity m(i, j) of the summand I (i, j) is an invariant of the persistence module.

17 Interval decomposition Theorem (Gabriel 970; Zomorodian & Carlsson 2002; etc.) Every (finite-dimensional) persistence module is isomorphic to a (finite) direct sum 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-generated graded modules over F[t].

18 Interval decomposition Theorem (Gabriel 970; Zomorodian & Carlsson 2002; etc.) Every (finite-dimensional) persistence module is isomorphic to a (finite) direct sum 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-generated graded modules over F[t]. Plot the multiplicities as points-with-multiplicity on a persistence diagram : m(i, j)

19 Relationship between the two diagrams Inverse transformations r(i, j) m(i, j)

20 Relationship between the two diagrams Inverse transformations r(i, j) 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 n p=0 q=j m(p, q)

21 Relationship between the two diagrams Inverse transformations r(i, j) m(i, j) rank diagram persistence diagram Each summand I (p, q) contributes to r(i, j) whenever p i j q. Thus: rank diagram persistence diagram The inverse transformation is r(i, j) = i n p=0 q=j m(p, q) m(i, j) = r(i, j) r(i, j) r(i, j + ) + r(i, j + ) by the inclusion exclusion principle.

22 Relationship between the two diagrams Inverse transformations r(i, j) m(i, j) rank diagram persistence diagram Each summand I (p, q) contributes to r(i, j) whenever p i j q. Thus: rank diagram persistence diagram The inverse transformation is r(i, j) = i n p=0 q=j m(p, q) m(i, j) = r(i, j) r(i, j) r(i, j + ) + r(i, j + ) by the inclusion exclusion principle.

23 Plotting the persistence diagram Plotting conventions For persistence modules such as the Vietoris Rips homology of a finite metric space, constructed with respect to a set of critical values r 0 < r < < r n we adopt the following convention. Each summand I (i, j) is drawn as a half-open interval [r i, r j+ ) in the barcode; or a point (r i, r j+ ) in the persistence diagram. We set r n+ = + by convention. Persistence diagram large scale large scale small scale small scale Persistence Persistence diagram diagram points Vin(b,d) de Silva Pomona College points (b,d) small small large large scale scale scale scale Barcode Barcode 5. Persistence intervals Diagrams intervals [b,d) [b,d)

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25 The ELZ Persistence Algorithm Edelsbrunner, Letscher, Zomorodian (2000) Given a nested sequence of simplicial complexes S 0 S... we wish to compute the persistence diagram of S n H d (S 0) H d (S )... H d (S n). We compute it for all d simultaneously (up to some maximum dimension).

26 The ELZ Persistence Algorithm Edelsbrunner, Letscher, Zomorodian (2000) Given a nested sequence of simplicial complexes S 0 S... we wish to compute the persistence diagram of S n H d (S 0) H d (S )... H d (S n). We compute it for all d simultaneously (up to some maximum dimension). Input A sequence of cells σ 0, σ,..., σ n. For each cell we are given: the dimension d = dim(σ k ); the boundary [σ k ], as a cycle built from existing (d )-cells. Output The persistence diagram for this nested sequence of formal cell-complexes.

27 The ELZ Persistence Algorithm Data structure We maintain an upper-triangular square matrix, size equal to the number of cells processed so far. Columns are coloured green, yellow, red. colour content leading coefficient green d-cycle α k yellow d-cycle β k red d-chain γ k non-zero Here d is the dimension of the cell σ k whose column it is. We maintain a bijection between the yellow and red columns. If yellow column k is paired with red column l, then k < l and Readout γ l = β k. Each green column j corresponds to an interval [j, + ). Each yellow red pair (k, l) corresponds to an interval [k, l).

28 The ELZ Persistence Algorithm Matrix Chains green : α 0 = [], α 2 = [3] yellow : β = [2] [] red : γ 3 = [, 2] Persistence intervals [0, + ), [, 3), [2, + )

29 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.

30 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.

31 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 of the associated red column to eliminate that leading term. Repeat.

32 The ELZ Persistence Algorithm Update Step Consider the next simplex σ k.

33 The ELZ Persistence Algorithm Update Step Consider the next simplex σ k. If [σ k ] was already a boundary: The cycle reduction lemma gives [σ k ] = ρ. We get a new green cycle α k = [σ k ] ρ.

34 The ELZ Persistence Algorithm Update Step Consider the next simplex σ k. If [σ k ] was already a boundary: The cycle reduction lemma gives [σ k ] = ρ. We get a new green cycle α k = [σ k ] ρ. If [σ k ] was not previously a boundary: The cycle reduction lemma gives [σ k ] = ρ + φ where the leading term of φ is c[σ j ] for some scalar c. Change the j-th column to yellow, and colour the k-th column red. Set β j = c φ, γ k = c ([σ k ] ρ) (discarding α j ). Pair the columns j and k.

35 The ELZ Persistence Algorithm Matrix Chains green : α 0 = [], α 2 = [3] yellow : β = [2] [] red : γ 3 = [, 2] Persistence intervals [0, + ), [, 3), [2, + )

36 The ELZ Persistence Algorithm Matrix Chains green : α 0 = [] yellow : β = [2] [], β 2 = [3] [2] red : γ 3 = [, 2], γ 4 = [2, 3] Persistence intervals [0, + ), [, 3), [2, 4)

37 The ELZ Persistence Algorithm Matrix Chains green : α 0 = [], α 5 = [, 3] [2, 3] [, 2] yellow : β = [2] [], β 2 = [3] [2] red : γ 3 = [, 2], γ 4 = [2, 3] Persistence intervals [0, + ), [, 3), [2, 4), [5, + )

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