persistent homology and the stability theorempersistent homology and the stabilitytheorem ulrich...

Persistent Homology

and the

Stability Theorem

Ulrich Bauer

February 19, 2018

TAGS – Linking Topology to Algebraic Geometry and Statistics

http://ulrich-bauer.org/persistence-talk-leipzig.pdf 1 / 31

What is persistent homology?

0.1 0.2 0.4 0.8δ

Persistent homology is the homology of a Bltration

A Bltration is a certain diagram K ∶ R→ Top R is the poset category of (R, ≤) A topological space Kt for each t ∈ R An inclusion map Ks Kt for each s ≤ t ∈ R

Consider homology with coe?cients in a Beld (often Z2)

H∗ ∶ Top→ Vect Persistent homology is a diagram M ∶ R→ Vect

(persistence module)

H∗ ∶ Top→ Vect

Persistent homology is a diagram M ∶ R→ Vect(persistence module)

Homology inference

Given: Bnite sample P ⊂ Ω of unknown shape Ω ⊂ Rd

Problem (Homology inference)

Determine the homologyH∗(Ω).

Problem (Homological reconstruction)

Construct a shape X with H∗(X) ≅ H∗(Ω).

approximate the shape by a thickening Bδ(P) covering Ω

Requires strong assumptions:

Homology inference

approximate the shape by a thickening Bδ(P) covering ΩRequires strong assumptions:

Homology reconstruction by thickening

Theorem (Niyogi, Smale,Weinberger 2006)

Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that

Bδ(P) coversΩ, and

δ <√3/20 reach(M).

Then H∗(M) ≅ H∗(B2δ(P)).

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

Homology inference using persistence

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

LetΩ ⊂ Rd. Let P ⊂ Ω, δ > 0 be such that

the inclusionsΩ Bδ(Ω) B2δ(Ω) preserve homology.

Then H∗(Ω) ≅ imH∗(Bδ(P) B2δ(P)).

0.2 0.4 0.6 0.8 1.0 1.2

Homological realization

This motivates the homological realization problem:

Problem

Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that

H∗(X) = imH∗(L K).

Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)

The homological realization problem isNP-hard, even inR3.

Problem

H∗(X) = imH∗(L K).

Problem

H∗(X) = imH∗(L K).

Problem

H∗(X) = imH∗(L K).

This is not always possible:

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

The homological realization problem isNP-hard, even inR3.http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31

Stability

Persistence and stability: the big picture

Data point cloud P ⊂ Rd

Geometry function f ∶ Rd → R

Topology topological spaces (Bltration) K ∶ R→ Top

Algebra vector spaces (persistence module) M ∶ R→ Vect

Combinatorics intervals (persistence barcode) B ∶ R→Mch

distance

sublevel sets

distance

sublevel sets

homology

distance

sublevel sets

homology

structure theorem

Stability of persistence barcodes for functions

Let ∥f − g∥∞ = δ. Then there exists amatching between the

intervals of the persistence barcodes of f and g such that

matched intervals have endpointswithin distance ≤ δ, and

unmatched intervals have length ≤ 2δ.

Interleavings

Let δ = ∥f − g∥∞. Write Ft = f −1(−∞, t] and Gt = g−1(−∞, t].

Then the sublevel set Bltrations F,G ∶ R→ Top are

δ-interleaved:

Ft Ft+δ Ft+2δ Ft+3δ

Gt Gt+δ Gt+2δ Gt+3δ

Applying homology (functor) preserves commutativity

persistent homology of f , g yields

δ-interleaved persistence modules R→ Vect

Interleavings

δ-interleaved:

Interleavings

δ-interleaved:

Geometric interleavings

Persistence modules

A persistence module M is a diagram (functor) R→ Vect:

a vector space Mt for each t ∈ R a linear map mt

s ∶ Ms →Mt for each s ≤ t (transition maps)

respecting identity: mtt = idMt

and composition: mts ms

r = mtr

If each dimMt <∞, we say M is pointwise Bnite dimensional.

Amorphism f ∶ M → N is a natural transformation:

a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:

Persistence modules

A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R

a linear map mts ∶ Ms →Mt for each s ≤ t (transition maps)

r = mtr

Persistence modules

A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R a linear map mt

r = mtr

Persistence modules

r = mtr

Persistence modules

r = mtr

Persistence modules

r = mtr

a linear map ft ∶ Mt → Nt for each t ∈ R

morphism and transition maps commute:

Persistence modules

r = mtr

Interval PersistenceModules

Let K be a Beld. For an arbitrary interval I ⊆ R,

deBne the interval persistence module K(I) by

K(I)t =

⎧⎪⎪⎨⎪⎪⎩

K if t ∈ I,0 otherwise,

with transition maps of maximal rank.

Schematic example:

0 0 K K K 0 0

Barcodes: the structure of persistence modules

Theorem (Krull–Schmidt; Crawley-Boewey 2015)

Let M be a pointwise Bnite-dimensional persistence module.

Then M is interval-decomposable:

there exists a unique collection of intervals B(M) such that

M ≅ ⊕I∈B(M)

B(M) is called the barcode of M.B(M)

The decomposition itself is not unique.

This is why we use homology with coe?cients in a Beld.

M ≅ ⊕I∈B(M)

Algebraic stability of persistence barcodes

Theorem (Chazal et al. 2009, 2012; B, Lesnick 2015)

If two persistence modules are δ-interleaved,

then there exists a δ-matching of their barcodes:

matched intervals have endpoints within distance ≤ δ,

Mt Mt+δ Mt+2δ

Nt Nt+δ Nt+2δ

Barcodes as diagrams

The matching category

Amatching σ ∶ S↛ T is a bijection S′ → T′, where S′ ⊆ S, T′ ⊆ T.

Composition of matchings σ ∶ S↛ T and τ ∶ T ↛ U :

Matchings form a categoryMch objects: sets

morphisms: matchings

Barcodes as matching diagrams

We can regard a barcode B as a functor R→Mch:

For each real number t, let Bt be the set of intervals of Bthat contain t, and

for each s ≤ t, deBne the matching Bs ↛ Bt

to be the identity on Bs ∩ Bt .

0.1 0.2 0.4 0.8∆

0.1 0.2 0.4 0.8δ

Stability via functoriality?

Ft Ft+2δ

Gt+δ Gt+3δ

H∗(Ft) H∗(Ft+2δ)

H∗(Gt+δ) H∗(Gt+3δ)

B(H∗(Ft)) B(H∗(Ft+2δ))

B(H∗(Gt+δ)) B(H∗(Gt+3δ))

B(H∗(Ft)) B(H∗(Ft+2δ))

B(H∗(Gt+δ)) B(H∗(Gt+3δ))

Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2015)

There exists no functorVectR →MchRsending each persistence

module to its barcode.

Proposition

There exists no functorVect→Mch sending each vector space of

dimension d to a set of cardinality d.

Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2015)

There exists no functorVectR →MchRsending each persistence

module to its barcode.

Proposition

There exists no functorVect→Mch sending each vector space of

dimension d to a set of cardinality d.

Induced barcode matchings

Structure of persistence submodules / quotients

Proposition

Let f ∶ M → N be amonomorphism of persistence modules:

each ft ∶ Mt → Nt is injective.

Then f induces an injective map B(M)→ B(N)

mapping each I ∈ B(M) to some J ∈ B(N)

with larger or same left and same right endpoint.

Dually for epimorphisms (left and right exchanged).

Induced matchings

For a general morphism f ∶ M → N of persistence modules:

consider epi-mono factorization

M ↠ im f N .

im f N induces injection B(im f ) B(N)

M ↠ im f induces injection B(im f ) B(M)

compose to amatching B(M)↛ B(N):

B(im f )

Induced matchings

For a general morphism f ∶ M → N of persistence modules:

consider epi-mono factorization

M ↠ im f N .

im f N induces injection B(im f ) B(N)

M ↠ im f induces injection B(im f ) B(M)

compose to amatching B(M)↛ B(N):

B(im f )

Stability from interleavings

Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

imnt−δ,t+δ im ft Nt+δ

Mt ↠ im ft

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(im f )B(N•+δ)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(im f )B(im n•–δ,•+δ)

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Sending persistence into

Hilbert space

Extending the TDA pipeline

Mapping barcodes into a Hilbert space?

desirable for (kernel-based) machine learning methods

and statistics

stability (Lipschitz continuity): important for reliable

predictions

inverse stability (bi-Lipschitz): avoid loss of information

Existing kernels for persistence diagrams:

stability bounds only for 1−Wasserstein distance

Lipschitz constants depend on bound on number and

range of bars

no bi-Lipschitz bounds known

Can we hope for something better?

and statistics

predictions

range of bars

and statistics

predictions

range of bars

No bi-Lipschitz feature maps for persistence

Theorem (B, Carrière 2018)

There is no bi-Lipschitzmap from the persistence diagrams

(with the interleaving or any p−Wasserstein distance)

into any Bnite-dimensionalHilbert space,

evenwhen restricting to bounded range or number of bars.

If therewas such a bi-Lipschitzmap into someHilbert space,

the ratio of the Lipschitz constantswould have to go to∞

togetherwith the bounds on number or range of bars.

No bi-Lipschitz feature maps for persistence

There is no bi-Lipschitzmap from the persistence diagrams

(with the interleaving or any p−Wasserstein distance)

into any Bnite-dimensionalHilbert space,

evenwhen restricting to bounded range or number of bars.

If therewas such a bi-Lipschitzmap into someHilbert space,

the ratio of the Lipschitz constantswould have to go to∞

togetherwith the bounds on number or range of bars.

History

When was persistent homology invented?

[Edelsbrunner/Letscher/Zomorodian 2000]

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

When was persistent homology invented Brst?

When was persistent homology invented Brst?Web Images More… Sign in

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Morse’s functional topology

Key aspects:

early precursor of persistence and spectral sequences

uses Vietoris homology with Beld coe?cients

applies to a broad class of functions on metric spaces

(not necessarily continuous)

inclusions of sublevel sets have Bnite rank homology

(q-tame persistent homology)

focus on controlled behavior in pathological cases

Motivation and application: minimal surfaces

(from Dierkes et al.: Minimal Surfaces, Springer 2010)

Existence of unstable minimal surfaces

Using persistent homology:

Number of є-persistent critical points (minimal surfaces)

is Bnite for any є > 0 Morse inequalities for є-persistent critical points

Theorem (Morse, Tompkins 1939)

There is aC1 curve bounding an unstable minimal surface

(an index 1 critical point of the area functional).

Thanks for your attention!

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