Isomorphism of persistence modules

We show that the persistence moduleV_Ψconstructed in Equation4.46is isomorphic to the persistence module

V: Hn(R1)

ι1∗

−→ · · · ιN∗−1

To show that the persistence moduleV_Ψ is isomorphic toV, we will show that both

VΨandVare isomorphic to a third persistence module VTot: Hn(Tot1) ι1_Tot −→Hn(Tot2) ι2_Tot −→ · · · ι N−1 Tot −−→ Hn(TotN),

where each Hn(Toti)is the homology of the double complex from Diagram4.3for pa-

rameterei, and ιi_Totis the morphism induced by maps of double complexes. We will

first show that VTot is isomorphic to the persistence module V. This first step cor-

responds to constructing an isomorphism Φi_Tot for eachei that make the right half of

Diagram4.47commute (Theorem9). We will then show thatV_Ψis isomorphic toVTot,

by constructing mapsΦi that make the left half of Diagram4.47commute (Theorem

10). H0(C•F1_n)⊕H1(C•F_n1₋₁) Hn(Tot1) Hn(R1) H0(C•F2n)⊕H1(C•F_n2₋₁) Hn(Tot2) Hn(R2) .. . ... ... H0(C•FnN)⊕H1(C•F_nN₋₁) Hn(TotN) Hn(RN) Ψ1 Φ1 Φ1_Tot ι1_Tot ι1∗ Ψ2 Φ2 Φ2_Tot ι2_Tot ι2∗ ΨN−1 ιN_Tot−1 ιN∗−1 ΦN ΦN_Tot (4.47)

Before we proceed with the proof, we provide a summary of the construction of the homology of a double complex. For a fixede_i parameter, consider the 0thpage of the

spectral sequence in Diagram4.48. .. . ... L v∈NV C2(Riv) L e∈NV C2(Rie) L v∈NV C1(Riv) L e∈NV C1(Rie) L v∈NV C0(Riv) L e∈NV C0(Rie) ∂ ∂ ∂ ei 2 ∂ ∂ ei 1 ∂ ei 0 (4.48) Let C•_n,• = M v∈N_V Cn(Riv)⊕ M e∈N_V Cn−1(Rie),

and letDn:Cn•,• →C_n•,−•1be defined by

Dn=∂+ (−1)nei_n−1.

One can check that Dn−1◦Dn = 0, and hence obtain the following chain complex,

called total complex.

Toti_• : · · · _−→D3

C₂•,• _−→D2

C•₁,• _−→D1

C₀•,• _−→D0

0

LetHn(Toti)denote the homology of the total complex. Note that a coset ofHn(Toti)

is represented by[a,b], wherea ∈ L v∈NV Cn(Riv), b ∈ L e∈NV Cn−1(Rie), ∂b = 0 and∂a = (−1)n−1ei_n−1b.

A coset [a,b] is trivial in Hn(Toti) if there exist pn+1 ∈

L

v∈NV

Cn+1(Riv) and qn ∈

L

e∈NV

Cn(Rie)such that∂qn=band∂pn+1+ (−1)n+1ein(qn) =a.

parametere_i. There exists an inclusion map from double complex associated with pa-

rameter ei to that of parameter ei+1, as illustrated in Diagram 4.50. Each horizontal

face of Diagram4.50corresponds to a double complex for a parameterei. The vertical

maps ιi_n’s and κi_n’s constitute the inclusion maps of double complexes. Such inclu-

sion of double complexes induces morphisms on the corresponding total complexes Toti_• →Toti_•+1, which induces morphismιi_Tot : Hn(Toti) → Hn(Toti+1). The morphism

ιi_Totcan be written explicitly as

ιTot([a,b]) = [ιi_n(a),κi_n−1(b)]. (4.49) L v∈N_V Cn(R1v) L e∈N_V Cn(R1e) L v∈N_V Cn−1(R1v) L e∈N_V Cn−1(R1e) L v∈NV Cn(R2v) L e∈NV Cn(R2e) L v∈NV Cn−1(R2v) L e∈NV Cn−1(R2e) L v∈N_V Cn(R3v) L e∈N_V Cn(R3e) L v∈NV Cn−1(R3v) ... L e∈NV Cn−1(R3e) ... .. . ... ∂ _ι1 n e1 n κ1n ∂ ι1_n−1 e1 n−1 ∂ _ι2 n e2 n κ2n ∂ ι2_n−1 e2 n−1 κ1_n−1 ∂ ι3n e3 n κ3n ∂ ι3_n−1 e3 n−1 κ2_n−1 κ3_n−1 (4.50)

Theorem 9. There exist isomorphisms Ψi

Tot : Hn(Toti) → Hn(Ri)such that the following

diagram commutes. Hn(Tot1) Hn(R1) Hn(Tot2) Hn(R2) .. . ... Hn(TotN) Hn(RN) Ψ1 Tot ι1_Tot ι1∗ Ψ2 Tot ι2_Tot ι2∗ ιN_Tot−1 ι∗N−1 ΨN Tot (4.51)

Proof. We first define isomorphismsΨi_Tot : Hn(Toti)→ Hn(Ri). For each parametere_i,

letj_ni : L

v∈NV

Cn(Riv)→Cn(Ri)be a collection of inclusion maps. DefineΨi_Totby

Ψi

Tot([x,y]) = [jin(x)]. (4.52)

One can check thatΨi_Totis well-defined and bijective (AppendixB.4).

We now show that Diagram4.51commutes. Given[x,y]∈ Hn(Toti), we have

ιi_∗◦Ψi_Tot[x,y] =ιi_∗[ji_n(x)] = [ιi◦ji_n(x)],

and

Ψi+1

Tot ◦ιiTot[x,y] =ΨToti+1[ιin(x),κni−1(y)] = [jni+1◦ιin(x)].

The following diagram commutes because all the maps involved are inclusion maps.

L v∈NV Cn(Riv) Cn(Ri) L v∈NV Cn(Riv+1) Cn(Ri+1) ji n ιin ι i ji+1 n

So we know thatιi_∗◦Ψi_Tot=Ψ_Toti+1◦ιi_Tot. Thus, Diagram4.51commutes.

Theorem 10. There exists an isomorphismΦi _{: H}

0(C•Fi_n)⊕H1(C•F_ni₋₁) → Hn(Toti)for

every parametereisuch that the following diagram commutes.

H0(C•F1_n)⊕H1(C•F_n1₋₁) Hn(Tot1) H0(C•F2_n)⊕H1(C•F_n2₋₁) Hn(Tot2) .. . ... H0(C•FnN)⊕H1(C•F_nN₋₁) Hn(TotN) Ψ1 Φ1 ι1_Tot Ψ2 Φ2 ι2_Tot ΨN−1 _ιN−1 Tot ΦN (4.53)

Proof. Note that Lemma12 already tells us that there exists an isomorphism between

H0(C•Fin)⊕H1(C•F_ni−1) and Hn(Toti)for parameter ei < K. For clarity, we will de-

fine isomorphismsΦi : H0(C•F_ni)⊕H1(C•Fi_n−1) → Hn(Toti)explicitly on vectors of

H0(C•Fin)andH1(C•F_ni₋₁).

For{hxi} ∈ H0(C•Fin), let

Φi_{({hxi}, 0) = [x, 0]. (4.54)}

To define the mapΦi onH1(C•Fi_n−1), recall the basisBi of H1(C•Fi_n−1)from Equation

4.41. We will defineΦi on each basisBi as the following. Given{hb∗i} ∈Bi_{, let}

Φi_(0,{hb∗_{i}) = [(−1)}n+1_Γi_{hb∗_i},b∗_{], (4.55)}

whereΓi_{is the map defined in Equation4.38. Extend this map linearly to H}

1(C•Fi_n−1).

Note that given({hxi},{hyi})∈ H0(C•Fni)⊕H1(C•Fi_n−1), the mapΦi is Φi_{({hxi},{hyi}) =Φ}i_{({hxi}, 0) +Φ}i_(0,{hyi}).

We now show that Diagram4.53commutes. It suffices to show that each square of Diagram4.53commutes. H0(C•F_ni)⊕H1(C•Fi_n−1) Hn(Toti) H0(C•Fin+1)⊕H1(C•Fi_n+₋1₁) Hn(Toti+1) Ψi Φi ιi_Tot Φi+1 (4.56)

We will show that the above diagram commutes for each vector of H0(C•Fin) and

H1(C•Fi_n−1).

Case 1: Given{hxi} ∈H0(C•Fni), we know that

ιi_Tot◦Φi({hxi}, 0) =ιi_Tot([x, 0]) = [ιi_nx, 0].

On the other hand, note that{hιi

nxi} ∈H0(C•F_ni+1), and Φi+1_◦Ψi_{({hxi}, 0) =Φ}i+1_({hιi

nxi}, 0) = [ιinx, 0].

Thus, the diagram commutes for every{hxi} ∈ H0(C•Fin).

Case 2: To show that the diagram commutes for every vector inH1(C•F_ni₋₁), it suffices

to show that the diagram commutes for the basis Bi of H1(C•Fi_n−1) from Equation

4.41. We consider two cases separately: the first, if{hb∗i} ∈ _Bi

ker, and the second, if

{hb∗i} ∈Bi A.

Case 2A: Assume{hb∗i} ∈_Bi

ker. We know that

On the other hand, Φi+1_◦Ψi_(0,{hb∗_{i}) =Φ}i+1₍₍₋₁₎n+1 ψi{hb∗i},{hκi_n−1b∗i}) = Φi+1((−1)n+1{h−e_ni+1αi+1+ιi_n◦Γi{hb∗i}i}, 0) = [−(−1)n+1ei_n+1αi+1+ (−1)n+1ιi_n◦Γi{hb∗i}, 0] Then, ιi_Tot◦Φi(0,{hb∗i})−Φi+1◦Ψi(0,{hb∗i}) = [(−1)n+1ei_n+1αi+1,κi_n−1b∗].

Recall from Equation 4.43that αi+1 ∈ L

e∈NV

Cn(Rie+1)satisfies κni−1b∗ = ∂αi+1. Thus,

ιi_Tot◦Φi(0,{hb∗i})−Φi+1◦Ψi(0,{hb∗i}) = 0, and the diagram commutes for basis

vectors{hb∗i} ∈Bi ker.

Case 2B: If{hb∗i} ∈Bi

A, then again,

ιi_Tot◦Φi(0,{hb∗i}) =ιi_Tot([(−1)n+1Γi{hb∗i},b∗]) = [(−1)n+1ιi_n◦Γi{hb∗i},κi_n−1b∗].

On the other hand,

Φi+1_◦Ψi_(0,{hb∗_{i}) =Φ}i+1₍

ψi{hb∗i},{hκi_n−1b∗i}) =Φi+1(0,{hκi_n−1b∗i})

= [(−1)n+1Γi+1{hκ_ni₋₁b∗i},κ_ni₋₁b∗] = [(−1)n+1ιi_n◦Γi{hb∗i},κ_ni₋₁b∗].

The third equality follows from the fact thatΓi+1was defined in Equation4.38such that

Γi+1_{hκi

n−1b∗i} = ιinΓi{hb∗i}. Thus, the diagram commutes for basis vectors{hb∗i} ∈

Bi A.

The following immediate corollary tells us that the persistence module

VΨ :H0(C•Fn1)⊕H1(C•F1n−1) Ψ

1

−→ · · ·−−−→ΨN−1 H0(C•FnN)⊕H1(C•FnN−1)

constructed via distributed computation is isomorphic to the persistence module

V: Hn(R1)

ι1∗

−→ · · · ιN∗−1

−−→Hn(RN.)

Hence,barcode(V_Ψ) =barcode(V).

Corollary 2. The mapsΦi _andΦi

Totare isomorphisms that make the Diagram4.47commute.

In §4.2.2, we defined the map ψi : H1(C•F_ni−1) → H0(C•Fin+1) in Equation 4.27

by extending a mapδi constructed in Equation 4.24. In §4.2.3, we redefined the map ψi : H1(C•Fi_n−1)→ H0(C•Fni+1)explicitly on a basisBiofH1(C•F_ni₋₁)in Equation4.43.

One might wonder why it was necessary for us to reconstruct the map ψi explicitly

when we already hadψi defined in Equation4.27.

To clarify the discussion, let

ψi :H1(C•Fi_n−1)→ H0(C•Fi_n+1)

denote the map defined in Equation4.27, and let

ψi_B: H1(C•Fin−1)→ H0(C•Fni+1)

denote the map constructed explicitly on a basis Bi in Equation 4.43. Let Ψi :

H0(C•Fin)⊕H1(C•Fi_n−1)→H0(C•Fin+1)⊕H1(C•F_ni+−11)be the map defined by

Ψi_{({hxi},{hyi}) = (H}

and letΨi_B: H0(C•Fi_n)⊕H1(C•Fi_n−1)→ H0(C•Fi_n+1)⊕H1(C•Fi_n+₋1₁)be the map defined

by

Ψi

B({hxi},{hyi}) = (H0(φi_n){hxi}+ (−1)n+1ψ_Bi {hyi},H1(φ_ni₋₁){hyi}). (4.58)

When given just two parameters, sayeiandei+1, then the two mapsΨiandΨi_Bboth

define persistence modules

VΨ :H0(C•F_ni)⊕H1(C•Fi_n−1) Ψi −→ H0(C•Fi_n+1)⊕H1(C•F_ni+₋1₁), VB_Ψ :H0(C•Fni)⊕H1(C•Fin−1) Ψi B −→ H0(C•Fni+1)⊕H1(C•Fi_n+−11)

that are each isomorphic to the persistence module

V: Hn(Ri)→ Hn(Ri+1).

However, given more than two parameters, the persistence module defined by the mapsΨi may not be isomorphic to the persistence module of interest. We provide an illustration of the disparity in the following example.

Example 15. Consider the following example point cloudP, a map f : P → R, and a coverVof f(P)in Figure4.10. The coverVconsists of two intervals,VBandVR. Assume

that we are given two parameterse1 < e2. The Rips complexes and the Rips systems

FIGURE4.10: A point cloudP, map f :P→R, and a coverV

(A) Rips system for parametere1 (B) Rips complex for parametere1

(C) Rips system for parametere2 (D) Rips complex for parametere2

FIGURE4.11: Rips systems and Rips complexes for parameterse1ande2

for computingH1(Ri)are

H0(C•F11) =K, H1(C•F10) =K

H0(C•F12) =K, H1(C•F20) =0.

The map H0(φ₁1) : H0(C•F₁1) → H0(C•F2₁) is the identity map, and the map

H1(φ1₀) : H1(C•F1₀) → H1(C•F₀2) is the trivial map. Then, we can construct a map

δ1 : kerH₁(φ1₀) → cokerH0(φ1₁)as we have in Equation 4.24. Note thatδ1 is a triv-

ial map since cokerH0(φ1₁)is trivial. Hence, when we extend δ1 to ψ1 as we have in

Equation4.27, we obtain a trivial map

ψ1 : H1(C•F1₀)→H0(C•F₀2).

When we define the map

Ψ1 _:H

as in Equation4.57, we obtain the following persistence module

V_Ψ1 :K⊕K Ψ 1

−→ K,

whereΨ1can be expressed by the matrix

1 0

.

The persistence moduleV_Ψ1 is illustrated in Figure4.12.

FIGURE4.12: Persistence moduleΨ1

One can check thatV_Ψ1 is isomorphic to the persistence module

V :H1(R1)→ H1(R2).

The persistence moduleVis illustrated in Figure4.13. The map between the left cycles represents the map H0(φ₁1) : H0(C•F1₁) → H0(C•F2₁), and the map between the right

cycles represents the map H1(φ1₀) : H1(C•F₀1) → H1(C•F2₁). In particular, we can see

thatH1(C•F₀1)represents the right cycle in Figure4.13.

Let’s now consider what happens if we were given three parameters e0,e1, ande2.

The Rips systems and the Rips complexes for the three parameters are illustrated in Figure4.14.

FIGURE4.13: Rips complexesR1,→R2

(A) Rips system for parametere0 (B) Rips complex for parametere0

(C) Rips system for parametere1 (D) Rips complex for parametere1

(E) Rips system for parametere2 (F) Rips complex for parametere2

The relevant cosheaf homologies for computingH1(Ri)are H0(C•F10) =0, H1(C•F00) =K H0(C•F11) =K, H1(C•F10) =K H0(C•F₁2) =K, H1(C•F20) =0. The map Ψ0 _{: H} 0(C•F0₁)⊕H1(C•F₀0) → H0(C•F₁1)⊕H1(C•F₀1) maps H1(C•F0₀)

identically to H1(C•F1₀). Note that the mapΨ1has been constructed in Equation4.59.

Then, we obtain a persistence module

VΨ:K Ψ

0

−→K⊕K−→Ψ1 K, whereΨ0is represented by the matrix

   0 1   ,

andΨ1is represented by the matrix

1 0

.

Figure4.15illustrates the persistence moduleV_Ψ.

FIGURE4.15: Persistence moduleV_Ψ

Note that this persistence module is not isomorphic to the persistence module

(A) Barcode forVΨ (B) Barcode forV FIGURE4.16: Barcodes for persistence modulesVΨandV

(A) Cycle represented by

{hyi}

(B) Cycle that should be rep-

resented byΨ0_(0,{hyi})

(C) Cycle that should be rep- resented byΨ1◦Ψ0_(0,{hyi}

FIGURE4.17: Cycles that should be represented at parameterse0,e1, ande2

The difference between the two persistence modulesV_Ψ andV are illustrated by the different barcodes in Figure4.16.

The disparity occurs because the mapΨ0_andΨ1_{effectively determines the cycle of}

R1_{represented byH}

1(C•F1₀). The construction of the mapsΨ0 andΨ1assume distinct

cycle representations ofH1(C•F1₀).

Let’s start with a basis element {hyi} ∈ H1(C•F0₀). This element{hyi}represents

the cycle illustrated in Figure4.17a. The imageΨ0(0,{hyi}) = (0,H1(φ0₀){hyi})must

represent the cycle illustrated in Figure 4.17b, and the image Ψ1◦Ψ0_{(0,{hyi})must}

represent the cycle illustrated in Figure4.17c.

In reality, we have seen in Figure 4.13 that H1(C•F₀1), and hence Ψ0(0,{hyi}) =

(0,H1(φ0₀){hyi}), represents a cycle illustrated in Figure 4.18. It is such inconsistency

in the represented cycles that preventsV_Ψ from being isomorphic to the persistence moduleV.

The construction of §4.2.3fixes this issue, by ensuring thatH0(C•F₀1)actually rep-

Chapter 5

Multiscale Persistent Homology

Data often comes with additional properties one might want to consider during the analysis process. For example, density estimate, coordinates, and time dependence of a point cloud are some of the factors that can affect one’s analysis of a barcode. The goal of this chapter is to introduce a multiscale analysis framework for persistent homology using the distributed computation method from Chapter4. The general structure for multiscale analysis is provided in §5.1. The result of this framework is a barcode an- notated with the properties of interest. One can then use this annotated barcode for a finer analysis taking the characteristics into account. For example, one may analyze any trends in the barcode or if the significant features share a common property. In §5.2, we study a dataset in which the feature sizes depend on the density of the constituting points. In such situations, the annotated barcode allows the user to detect significant features that are overlooked by standard persistent homology methods.

5.1

Multiscale Persistent Homology

We provide a general schematic for using distributed persistent homology computation for multiscale analysis purposes. Our goal is to enrich the barcode so that it reflects properties of interest. In particular, given a point cloud P, let f : P → R be a map that reflects some characteristic of the point cloud. For example, f can be a projection map to one of the coordinates, a density estimate, distance to a landmark, or any other characteristic of interest. Construct a coverVof f(P)so that points p∈ Pwith similar

f(p)values belong to the same member ofV. In particular, choose a coverVsuch that its nerveN_Vis a compact subset ofR.

LetVdenote the persistence module

V:Hn(R1)→ Hn(R2)→ · · · → Hn(RN)

obtained by applying persistence to entire point cloud P in the usual sense. Let

barcode(V) denote the barcode of V. Recall that each bar of a barcode represents a feature in the Rips complexes. If a bar with birth time ei represents a feature γ that

consists of points in f−1_{(U)for someU ∈ V, we say that the featureγlives inU, and}

we annotate the corresponding bar with the setU. Our goal is to annotate the bars of

barcode(V)by such setsUofV.

An algorithmic summary of the annotation process is provided, followed by a de- tailed explanation of each step.

Algorithm 1Annotatebarcode(V).

1: ComputeV∗ using distributed computation.

2: Label vector spaces ofV∗.

3: For each persistence moduleWsofV∗ =L

s Ws, annotate

barcode(W_s).

4: From the annotatedbarcode(V∗), annotatebarcode(V).

5: Return annotatedbarcode(V).

Step 1. Compute persistence moduleV∗

Recall thatVis the persistence module

V:Hn(R1)→ Hn(R2)→ · · · → Hn(RN)

of interest. LeteLbe the largesteparameter such that

i.e.,eL <K, whereKis the upper bound from Lemma13. LetV|Ldenote the sequence

of vector spaces and maps ofVup to parametereL:

V|_L :Hn(R1)→ Hn(R2)→ · · · → Hn(RL).

We can compute the persistence module

V|Ψ_L : H0(C•F1n)⊕H1(C•F1n−1) Ψ

1

−→. . .−−→ΨL−1 H0(C•FnL)⊕H1(C•FnL−1), (5.1)

that is isomorphic toV|_Lusing the distributed computation method from Chapter 4. Each mapΨi is defined in Equation4.45.

In fact, instead of computing the persistence moduleV|Ψ_L, we will compute a persis- tence moduleV∗ that is isomorphic toV|Ψ_L, and hence isomorphic toV|L, that can re-

veal some additional information about the features represented by the barcode. Recall from §2.2.5that for each parameterei, the cosheafFincan be decomposed asFin∼= ⊕Iin,

where eachIi

n is an indecomposable cosheaf over NV. In other words, there exists an

isomorphism of cosheaves

Di_n:Fi_n→ ⊕Ii_n. (5.2) For each parameterei, letVi∗denote the vector space

Vi

∗ = H0(C•⊕Ii_n)⊕H1(C•Fi_n−1).

Let(V|Ψ_L)i _{denote thei}th_{vector space of the persistence moduleV|}Ψ

L defined in Equa-

tion5.1. Note the difference betweenVi_∗and(V|Ψ_L)i: we only replaced the cosheafF_ni by the direct sum of its indecomposables. Cosheaf F_ni₋₁ remains intact. The isomor- phismDi_nof cosheaves from Equation5.2induces an isomorphismαi : (V|Ψ_L)i → Vi∗

defined by

where H0(Din) : H0(C•F_ni) → H0(C•⊕Ii_n)is the map induced by D_ni. LetV∗ be the persistence module V∗ :H0(C•⊕I1_n)⊕H1(C•F1_n−1) Ψ1 ∗ −→ · · · ΨN∗−1 −−−→H0(C•⊕IN_n)⊕H1(C•F_nN−1),

where the mapΨi∗ is defined byΨi∗ =αi+1◦Ψi◦(αi)−1.

To write Ψi

∗ explicitly, recall from Equation 4.45 that given a pair of parameters

ei < ei+1 < K, the mapΨi : H0(C•Fni)⊕H1(C•F_ni₋₁) → H0(C•Fin+1)⊕H1(C•Fi_n+₋1₁)is

defined by

Ψi_{({hxi},{hyi}) = (H}

0(φi_n){hxi}+ (−1)n+1ψi{hyi}, H1(φ_ni₋₁){hyi}),

where H0(φi_n)and H1(φi_n−1)are the maps defined in Equation4.16andψi is the map

defined in Equation4.43. Then, Ψi∗ : H0(C•⊕Iin)⊕H1(C•Fi_n−1) → H0(C•⊕Iin+1)⊕

H1(C•Fi_n+−11)is defined by

Ψi

∗({hxi},{hyi}) = (H0(φ_ni_∗){hxi}+ (−1)n+1ψi_∗{hyi}, H1(φi_n−1){hyi}), (5.3)

whereH0(φi_n∗)is the map induced by

φi_n∗ = Dni+1◦φin◦(Dni)−1 (5.4)

In document Cellular Sheaves And Cosheaves For Distributed Topological Data Analysis (Page 109-132)