Compatible basis, Stability of persistent homology, and Homology inference

K0=σ1K1σ2K2σ3σnKn={σ1,,σn}Ki1σiKi=Ki1{σi}

Compatible basis

Compatible basis at Kn:
  • cycles c1,,cn𝒞(Kn)
  • a partition {1,,n}=FGH

  • bijection GH, denoted by 𝒫={(g,h)}

such that

  1. i,c1,,ci=𝒞(Ki)ci=ε1/2σ1++εi1σi1+σi0 in /2
  2. fF,cf=0
  3. (g,h)𝒫,ch=cgcg=0

Properties:

1.

i{1,,n},{cf}fi,fF{cg}gi,gG generate Z(Ki)

Consider the restriction map

R:{𝒞(Kn)𝒞(Ki)σσ if σKiτ0 if τKnKicj(ji)cj

cf. picture

2.

i,{cg}(g,h)𝒫,g<hi generate B(Ki)

3.

{[cf]}fF,fi{[cg]}(g,h)𝒫,gi<h generate H(Ki)

Example: cf. picture

You can basically read the decomposition of the persistence module from a compatible basis.

Th:

Let K0Kn and a compatible basis {c1,,cn}, FGH={1,,n}, 𝒫G×H

Then the persitence module is:

(Kn)fF𝕀[f;n](g,h)𝒫𝕀[g;h1]

cf. picture

Sketch of proof:

Consider one ci:

𝒞(K0)𝒞(K1)𝒞(Kn)

(cf picture)

Iterative algorithm

Suppose we have a compatible basis for the filtration

K0=σ1K1σ2K2σ3σiKi

{c1,,cn}, FGH={1,,n}, 𝒫G×H

Consider the filtration

σ1K1Kiσi+1Ki+1

update the compatible basis to Ki+1.

Similar to what we did during the previous class: creator or destructor?

  1. Creator: if [σi+1]=0 in H(Ki), then σi+1B(Ki)

    Consequently:

    σi+1=gGεgcg=(g,h)𝒫εgch(σi+1(g,h)𝒫εgch)=0

    So ci+1 is a cycle:

    ci+1=σi+1(g,h)𝒫εgchZ(Ki+1)
    • ci+1 is independent of the other cj’s in the compatible basis:
    c1,,ci+1=𝒞(Ki+1)
    • and ci+1 cannot be the boundary of another chain because σi+1 has no coface.

    Set

    FF{i+1}

    compatible basis is c1,,ci,ci+1

  2. Destructor: if [σi+1]0 in H(Ki), then

    σi+1=gGεgcg+fFεfcf0

    since

    {cf}fi,fF{cg}gi,gG generate Z(Ki)

    Pick

    f0max{fFεf0}

    As

    (σi+1(g,h)𝒫εgch)=fFεfcf

    we can set

    cf0fFεfcf leading term remains σf0ci+1σi+1(g,h)𝒫εgch leading term remains σi+1

    (Condition 1 of the compatible bases remains true)

    Now, pair f0 and i+1:

    HH{i+1}FF{f0}GG{f0}𝒫𝒫{(f0,i+1)}

    (like last time, we “kill the youngest” (f0))

Example: cf. picture

NB: picking f0 to be the max amounts to killing off (i.e. putting it in G ⇒ trivial homology) the “youngest” (most recent) ci in the persistence module (cf. last week)

Matrix Implementation

We don’t really need to keep track of H actually.

Let M be the matrix of

:𝒞(Kn)=dCd(Kn)𝒞(Kn)

in the canonical basis {σ1,,σn} (boundary map extended to all dimensions, convenient way to only deal with one matrix).

Define low(i) to be the index of the lowest non-zero coefficient in the i-th column coli of M.

If coli=0, then low(i) is undefined.

Algorithm

for i = 1  n:
    while  j < i s.t. low(j) = low(i):
        col_i  col_i + col_j (in /2) # amounts to writing ∂σ_{i+1}  = ∑_{g∈G} ε_g c_g + ∑_{f∈F} ε_f c_f
    if col_i  0: # you have c_f's
        add (low(i), i-1) to the persistence diagram # what we called (f_0 = g, h-1) before, we pair f_0 = low(i) with index i

# Tackle the (f, n)
Add all (i, n) s.t. col_i = 0 and there is no column col_j with low(j) = i to the persistence diagram

The algorithm maintains the following property: at the end of the i-th iteration of the for loop, the matrix of looks like this: cf. picture

Stability of persistent homology

Let

D={(bi,di)}iI{(x,x)x}D={(bj,dj)}jJ{(x,x)x}

be two persistence diagrams.

To compare them, we will introduce a notion of “distance” between them.

The bottleneck distance between D and D:

is defined as

dB(D,D)infϕ:DDsuppDpϕ(p)

Note that bijections ϕ exist thanks to the addition of the diagonal (that’s the only reason we added the diagonal to persistence diagrams in the first place).

cf. picture

How are the original of geometry and the one of the persistence diagrams related?

Stability theorem

Th (Stability): Let f,g:K, where K is a finite simplicial complex, such that f and g induce a level-set filtration on K.

Kα=f1(,α] is a simplicial complex

Let Dgm(f) be the persistence diagram of the filtration

Kα1Kα2Kαn

where the αi’s are in the image of f.

Define similarly Dgm(g). Then

dB(Dgm(f),Dgm(g))fg=maxσK|f(σ)g(σ)|

Continuous version:

Th (Stability 2): Let f,g: where is a metric space, such that f and g are tame.

f and g define level-set filtrations

{f1(,r]}r and {g1(,r]}r

then

dB(Dgm(f),Dgm(g))fg

Philosophy of the proof:

fgε

so

r,f1(,r]g1(,r+ε]f1(,r+2ε]

cf. picture

Application: Homology inference

Back to our very first motivation: we’re given a set of points, how do we unravel their underlying topology?

Reminder:

Haussdorff distance between compact sets X,YN:
dH(X,Y)=max{supxXd(x,Y),supyYd(y,X)}

where d(x,Y)=infyYd(x,Y)

(cf. picture)

A set of points P is an ε-sample of a compact set KN:

if dH(P,K)ε

Note that for every compact set KN induces a “distance-to-this-compact” function

dK:{Nxd(x,K)

Level-set

d1(,r]={ all points of N that are at distance at most r from K}

Measure of “complexity” of the geometry of a compact

Let K be a compact.

A critical point for dK:

is a point xN with at least two nearest neighbors on K. I.e.

y1,y2K,y1y2 s.t. d(x,y1)=d(x,y2)=dK(x)
The reach of K:

is reach(K)inf{dK(x)x is critical}

Inference theorem

Let KN be a “nice” compact set of reach ε>0.

Let P be a ν-sample of K, with ν«ε.

Then there is a one-to-one correspondence between the points of the persistence diagram of the Cech filtration on P that are ν-away from the diagonal and the homology features of K.

cf. picture

Sketch of proof:

Claims:

  1. dK1(,r] has the same homotopy type as K for all r<reach(K)

    (cf. picture)

  2. if P is a ν-sample, i.e. dH(P,K)ν, then

    dPdKν
  3. Stability theorem: dB(Dgm(dK),Dgm(dP))ν=dKdP

  4. dP1(,r]=pPB(p,r)Nerve lemma𝒞Cechr(P)
  5. Persistent homolgy of dK will be expressed as something like 𝕀[0,ε]

cf. picture

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