Construction of distributed persistence module

Given a family of parameters(ei)iN=1such that the isomorphism

Hn(Ri)∼= H0(C•Fin)⊕H1(C•Fin−1)

holds for eachei, we now construct maps

Ψi _{: H}

0(C•Fin)⊕H1(C•F_ni−1)→H0(C•Fin+1)⊕H1(C•F_ni+₋1₁)

such that the resulting persistence module

VΨ :H0(C•Fn1)⊕H1(C•F1n−1) Ψ

1

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

is isomorphic to the persistence module

V: Hn(R1) ι1∗ −→ · · · ιN −1 ∗ −−→Hn(RN) from Equation4.14.

Recall the maps

H0(φi_n):H0(C•Fni)→H0(C•Fni+1)

H1(φi_n−1):H1(C•F_ni−1)→H1(C•F_ni+₋1₁)

(4.29)

from Equation4.16and the map

ψi :H1(C•Fin−1)→ H0(C•Fin+1)

from Equation4.27. For eachei, it is possible to construct a map

Ψi _{: H}

0(C•Fi_n)⊕H1(C•F_ni−1)→H0(C•Fi_n+1)⊕H1(C•F_ni+₋1₁)

by

Ψi_{({hxi},{hyi}) = (H}

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

Given two parametersei <ei+1, the persistence module

H0(C•Fni)⊕H1(C•Fin−1) Ψ

i

−→H0(C•Fni+1)⊕H1(C•F_ni+₋1₁)

can be shown to be isomorphic to the persistence module

Hn(Ri)

ιi∗

−→ Hn(Ri+1).

Given more than two parameters, the mapsΨi_{may define a persistence module that}

is different from the persistence moduleVthat we are interested in. Thus, one should be mindful of the subtleties involved when constructing a persistence module for more than two parameters. Example15at the end of this chapter illustrates a situation where the maps Ψi _{lead to a persistence module that is not isomorphic to the persistence}

module of interest.

The subtlety arises from the fact that for each parameterei, the cosheaf homology

multiple e_i parameters, the naive construction might force H1(C•Fi_n−1) to represent

cycles that are not compatible with other parameters.

In this section, we construct morphismsΨi that ensures thatH1(C•Fi_n−1)represent

cycles that are compatible across parameters. We achieve this goal by reconstructing morphisms

ψi :H1(C•Fi_n−1)→H0(C•Fin+1).

Recall from Equations4.24and4.27that a mapψi : H1(C•F_ni₋₁)→H0(C•F_ni+1)has

been defined by extending the mapδi. The mapδiwas defined by

δi[{hγi}] = [{h−ei_n+1αi+1+ιi_nβii}],

whereαi+1satisfies Equation4.22andβisatisfies Equation4.23.

Instead of extending the map δi, we will define the map ψi : H1(C•F_ni₋₁) →

H0(C•Fin+1)directly, as

ψi{hγi}={h−e_ni+1αi+1+ιi_nβi}

for{hγi} ∈kerH1(φi_n−1).

When there are multiple candidates for αi+1 and βi, the different choices of αi+1

and βi can lead to different constructions of ψi. It turns out, the different choices for αi+1do not affect the map ψi. It is the different choices ofβi that can lead to varying

constructions ofψi. In particular, we want to choose elementsβi such that the mapsψi

are compatible across parameters.

To that end, we construct the maps ψi on a fixed basisBi of H1(C•F_ni₋₁)in an in-

ductive manner so that we can guarantee that choices ofβifor a parametereiare com-

patible with the choices ofβi−1for parameterei−1. Once we define the mapψi, we will

be able to defineΨi.

The construction of map ψi will make use of Diagram 4.30. Note that while the

previous commutative diagrams were constructed for two parameters ei < ei+1, the

the following diagram is seen from a different perspective in three dimensions than the previous diagrams for display purposes. In this diagram, each horizontal face of the cube corresponds to the 0thpage of the spectral sequence from Diagram4.3for a fixed

eparameter.

Note that ∂ are the usual boundary maps and ei_n : L

e∈NV

Cn(Rie) →

L

v∈NV

Cn(Riv)

is the inclusion of n-chains on the edges of N_V to the vertices of N_V. The map

ιi_n: L v∈N_V Cn(Riv)→ L v∈N_V Cn(Riv+1)andκin: L e∈N_V Cn(Rie)→ L e∈N_V

Cn(Rie+1)are both inclu-

sion maps. 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 (4.30)

For parametere1, we will fix a basisB1ofH1(C•F1_n−1). We will define a linear map

Γ1 _{: H}

1(C•F_n1₋₁) → L v∈NV

Cn(R1v)and a linear mapψ1 : H1(C•F1_n−1) → H0(C•F2_n)by

For each parametere_i, we will go through the following steps to construct the map ψi.

• Step 1. Fix a basis Ci of H1(C•Fi_n−1) that is compatible with the basis Bi−1 of

H1(C•Fi_n−₋1₁).

• Step 2. Define a mapΓi _{: H}

1(C•F_n1₋₁)→ L v∈NV

Cn(Riv)by defining the map onCi

and extending linearly.

• Step 3.Fix a new basisBi ofH1(C•Fi_n−1).

• Step 4.Define the mapψi : H1(C•Fi_n−1)→ H0(C•Fi_n+1)on basisBiusing the map Γi _{defined in Step 2.}

The mapΓi :H1(C•Fi_n−1)→ L v∈NV

(Ri_v)defined in Step 2 will encode the choice ofβi

for each basis vector{hbi} ∈Bi_{. The basisC}i _ofH

1(C•Fi_n−1)of Step 1 will allow us to

define the mapΓi in a manner compatible with mapΓi−1from the previous parameter

ei−1.

We proceed with the base case for parametere1.

Base case

Recall from Equation4.25that

H1(C•F1_n−1) =A1⊕kerH1(φ1_n−1).

LetB1_Abe a basis ofA1_{, and letB}1

kerbe a basis of kerH1(φ1n−1). Then,

B1 _=B1

A∪B1ker

is a basis ofH1(C•F1_n−1). For each basis vector{hbi} ∈B1, fix a coset representativeb∗

ofhb∗i. Thus, we can express the basis as

B1_{= { {hb}∗

We will defineΓ1 : H1(C•F_n1₋₁) → L v∈NV

Cn(R1v)on the basisB1. For a basis vector

{hb∗i} ∈B1_{, we know from Equation4.23that there exists someβ}1_∈ L

v∈NV

Cn(R1v)such

that

∂β1= e1_n−1b∗.

Among all possible candidates for β1that satisfy the above equation, choose any par-

ticularβ1, and denote the chosen element byβ1∗. For each{hb∗i} ∈B1, let Γ1_{hb∗_i}=

β1∗. (4.31)

Extend this mapΓ1linearly to the vector spaceH1(C•F1_n−1), i.e., define Γ1_(a

1{hb1∗i}+· · ·+am{hb∗mi}) =a1Γ1{hb∗1i}+· · ·+amΓ1{hb∗mi}.

We now defineψ1 : H1(C•F_n1₋₁) → H0(C•F_n2)on the basisB1and extend the map

linearly. By construction, a basis vector{hb∗i} ∈ B1 _{must satisfy either{hb}∗_{i} ∈ B}1 ker

or {hb∗i} ∈ B1

A. If {hb∗i} ∈ B1ker, recall from Equation4.22that there exists anα2 ∈

L

e∈NV

Cn(R2e)such that

∂α2=κ1_n−1b∗. (4.32)

Defineψ1for each{hb∗i} ∈B1by ψ1{hb∗i}=        {h −e2_nα2+ι1_n◦Γ1{hb∗i} i} if{hb∗i} ∈B1_ker 0 if{hb∗i} ∈B1 A (4.33) whereα2 ∈ L e∈NV

Cn(R2e)can be any element satisfying Equation4.32andΓ1is the map

defined in Equation4.31.

Extend the mapψ1to the vector spaceH1(C•F_n1₋₁), i.e., let

One can check that the mapψ1is well-defined (AppendixB.2).

Inductive step

Recall from Equation4.25that

H1(C•F_ni−−11) = Ai

−1_⊕kerH

1(φi_n−₋1₁).

Assume that we are given a basisBi−1ofH1(C•F_ni−₋1₁)that has the form

Bi−1_=Bi−1

A ∪Biker−1,

whereBi_A−1 is a basis of Ai−1 andB_keri−1 is a basis of kerH1(φ_ni−₋1₁). Moreover, assume

that for each basis vector{hbi} ∈ Bi−1_{, a coset representativeb}∗_{of hbihas been fixed,}

i.e., we can write the basisBi−1as

Bi−1_{= { {hb}∗

1i}, . . . ,{hbi∗−1mi} }.

Lastly, assume that the mapΓi−1: H1(C•Fi_n−₋1₁)→ L v∈N_V

Cn(Riv−1)has been defined such

that for every{hb∗i} ∈Bi−1_{, we have}

∂Γi−1{hb∗i}=ei_n−₋1₁b∗. (4.35)

• Step 1. We first fix a basisCi of H1(C•Fi_n−1). By assumption, the basis Bi−1 of

H1(C•Fi_n−₋1₁)has the formBi−1= Bi_A−1∪Bi_ker−1. Without loss of generality, assume

that

Bi−1

A ={ {hb

∗

1i}, . . . ,{hb∗ti} }.

One can show that {hκi_n−₋1₁b₁∗i}, . . . ,{hκ_ni−₋1₁b∗_ti} are linearly independent in

H1(C•Fi_n−1)(AppendixB.3). Let Ci im={ {hκin−−11b ∗ 1i}, . . . ,{hκin−−11b ∗ ti} }.

ExtendCi_imto a basisCiof H1(C•Fi_n−1). LetCi_D denote the basis vectors ofCi that

are not inCi_im, i.e.,

Ci _=Ci

im∪CiD. (4.36)

For each{hci} ∈Ci_{, fix a coset representativec}∗_{ofhcias the following. If{hci} ∈}

Ci

imsuch that{hci}= {hκin−−11b∗si}, then letc∗ =κni−−11b∗s be the coset representative

ofhci. If{hci} ∈_Ci

D, fix any coset representativec∗ofhci.

• Step 2. We will now define Γi : H1(C•F_ni−1) →

L

v∈NV

Cn(Riv) on the basis Ci.

Given a basis vector {hc∗i} ∈ _Ci_{, we know from Equation4.23that there exists}

someβi ∈ L

v∈NV

Cn(Riv)such that

∂βi =ei_n−1c∗. (4.37)

If{hc∗i} = {hκi_n−₋1₁b∗i} ∈ Ci

im, then by Equation4.35and commutativity of Di-

agram 4.30, one can check that ιi_n−1Γi−1{hb∗i}is a candidate for βi that satisfies

Equation4.37. DefineΓion each{hc∗i} ∈Ci_by

Γi_{hc∗_i}=        ιi_n−1Γi−1{hb∗i} if{hc∗i}={hκi_n−₋1₁b∗i} ∈Ci_im βi if{hc∗i} ∈Ci_D (4.38) where βi ∈ L v∈NV

Cn(Riv)is any element satisfying Equation 4.37. Note that by

construction, for each{hc∗i} ∈_Ci_{, we definedΓ}i_c∗ _{such that}

∂Γi{hc∗i}=ei_n−1c∗. (4.39)

Extend the mapΓilinearly to the vector spaceH1(C•F_ni−1).

• Step 3.Recall from Equation4.25the decomposition

LetBi_Abe a basis ofAi_{, and letB}i

kerbe a basis of kerH1(φin−1). Then,

Bi _=Bi

A∪Bkeri

is a basis ofH1(C•F_ni−1). Each basis vector{hbi} ∈ Bi can be written as a linear

combination of basisCi ={ {hc∗₁i}, . . . ,{hc∗_li} }as

{hbi}=d1{hc1∗i}+· · ·+dl{hc∗li}.

Then, let

b∗ =d1c∗1+· · ·+dlc∗l (4.40)

be the coset representative ofhbi. We can express the basisBiof H1(C•Fi_n−1)as

Bi _{={ {hb}∗

1i}, . . . ,{hb∗ki} }. (4.41)

Note that by construction, for any basis vector{hb∗i} ∈Bi_{, we have}

Γi_{hb∗_i}=d

1Γi{hc∗1i}+· · ·+dlΓi{hc∗li}

and

∂Γi{hb∗i}=ei_n−1b∗,

which is a condition we need to pass on to the next inductive step.

• Step 4. We now defineψi : H1(C•F_ni−1) → H0(C•Fin+1)by defining ψi on the

basisBi and extending the map linearly. If {hb∗i} ∈ Bi

ker, recall from Equation

4.22that there exists someαi+1 ∈ L

e∈NV

Cn(Rie+1)such that

Defineψion each{hb∗i} ∈Bi by ψi{hb∗i}=        {h −ei_nαi+1+ιi_n−1◦Γi{hb∗i} i} if{hb∗i} ∈Bi_ker 0 if{hb∗i} ∈Bi A , (4.43) whereαi+1 ∈ L e∈N_V

Cn(Rie+1)is any element satisfying Equation4.42andΓi is the

map defined in Equation4.38. Define the linear mapψi by extending the above

to the vector spaceH1(C•F_ni₋₁)by

ψi(a1{hb∗1i}+· · ·+al{hb∗li}) =a1ψi{hb∗₁i}+· · ·+alψi{hb∗_li}. (4.44)

One can check thatψiis well-defined (AppendixB.2).

Once we define the mapsψi : H1(C•Fi_n−1)→ H0(C•Fi_n+1)inductively, we can define Ψi _{: H}

0(C•Fin)⊕H1(C•Fni−1)→H0(C•Fin+1)⊕H1(C•F_ni+−11)

by

Ψi_{({hxi},{hyi}) = (H}

0(φi_n){hxi}+ (−1)n+1ψi{hyi}, H1(φi_n−1){hyi}), (4.45)

whereH0(φi_n)andH1(φ_ni₋₁)are the maps defined in Equation4.16.

LetV_Ψdenote the persistence module

VΨ :H0(C•F1n)⊕H1(C•F1_n−1)

Ψ1

−→ · · ·−−−→ΨN−1 H0(C•FnN)⊕H1(C•F_nN−1). (4.46)

In document Cellular Sheaves And Cosheaves For Distributed Topological Data Analysis (Page 100-109)