Induced matchings and the a lge b r a i c st ab ilit y of persisten c - - PowerPoint PPT Presentation

Induced matchings and the algebraic stability

• f persistence barcodes

Ulrich Bauer

TUM

Apr 7, 2015 GETCO 2015, Aalborg

Joint work with Michael Lesnick (IMA)

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What is persistent homology?

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What is persistent homology?

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What is persistent homology?

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What is persistent homology?

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What is persistent homology?

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What is persistent homology?

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What is persistent homology?

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What is persistent homology?

Persistent homology is the homology of a filtration.

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What is persistent homology?

Persistent homology is the homology of a filtration.

• A filtration is a certain diagram K ∶ R → Top.

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What is persistent homology?

Persistent homology is the homology of a filtration.

• A filtration is a certain diagram K ∶ R → Top.
• A topological space Kt for each t ∈ R

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What is persistent homology?

Persistent homology is the homology of a filtration.

• A filtration is a certain diagram K ∶ R → Top.
• A topological space Kt for each t ∈ R
• An inclusion map Ks ↪ Kt for each s ≤ t ∈ R

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What is persistent homology?

Persistent homology is the homology of a filtration.

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

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Homology inference using persistent homology

Pδ = Bδ(P): δ-neighborhood (union of balls) around P

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

Let Ω ⊂ Rd. Let P ⊂ Ω be such that

• Ω ⊆ Pδ for some δ > 0 and
• both H∗(Ω ↪ Ωδ) and H∗(Ωδ ↪ Ω2δ) are isomorphisms.

Then H∗(Ω) ≅ imH∗(Pδ ↪ P2δ).

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Homology inference using persistent homology

Pδ = Bδ(P): δ-neighborhood (union of balls) around P

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

Let Ω ⊂ Rd. Let P ⊂ Ω be such that

• Ω ⊆ Pδ for some δ > 0 and
• both H∗(Ω ↪ Ωδ) and H∗(Ωδ ↪ Ω2δ) are isomorphisms.

Then H∗(Ω) ≅ imH∗(Pδ ↪ P2δ).

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Homology inference using persistent homology

Pδ = Bδ(P): δ-neighborhood (union of balls) around P

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

Let Ω ⊂ Rd. Let P ⊂ Ω be such that

• Ω ⊆ Pδ for some δ > 0 and
• both H∗(Ω ↪ Ωδ) and H∗(Ωδ ↪ Ω2δ) are isomorphisms.

Then H∗(Ω) ≅ imH∗(Pδ ↪ P2δ).

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Homology inference using persistent homology

Pδ = Bδ(P): δ-neighborhood (union of balls) around P

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

Let Ω ⊂ Rd. Let P ⊂ Ω be such that

• Ω ⊆ Pδ for some δ > 0 and
• both H∗(Ω ↪ Ωδ) and H∗(Ωδ ↪ Ω2δ) are isomorphisms.

Then H∗(Ω) ≅ imH∗(Pδ ↪ P2δ).

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The pipeline of topological data analysis

point cloud function topological spaces vector spaces intervals

distance sublevel sets homology barcode

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The pipeline of topological data analysis

point cloud

P ⊂ Rd

function topological spaces vector spaces intervals

distance sublevel sets homology barcode

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The pipeline of topological data analysis

point cloud

P ⊂ Rd

function topological spaces vector spaces intervals

distance x↦d(x,P) sublevel sets homology barcode

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The pipeline of topological data analysis

point cloud

P ⊂ Rd

function

f ∶ Rd → R

topological spaces vector spaces intervals

distance x↦d(x,P) sublevel sets homology barcode

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The pipeline of topological data analysis

point cloud function

f ∶ Rd → R

topological spaces vector spaces intervals

distance sublevel sets homology barcode

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The pipeline of topological data analysis

point cloud function

f ∶ Rd → R

topological spaces vector spaces intervals

distance sublevel sets t↦f −1(−∞,t] homology barcode

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The pipeline of topological data analysis

point cloud function

f ∶ Rd → R

topological spaces

K ∶ R → Top

vector spaces intervals

distance sublevel sets t↦f −1(−∞,t] homology barcode

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The pipeline of topological data analysis

point cloud function topological spaces

K ∶ R → Top

vector spaces intervals

distance sublevel sets homology barcode

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The pipeline of topological data analysis

point cloud function topological spaces

K ∶ R → Top

vector spaces intervals

distance sublevel sets homology t↦H∗(Kt;F) barcode

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The pipeline of topological data analysis

point cloud function topological spaces

K ∶ R → Top

vector spaces

M ∶ R → Vect

intervals

distance sublevel sets homology t↦H∗(Kt;F) barcode

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The pipeline of topological data analysis

point cloud function topological spaces vector spaces

M ∶ R → Vect

intervals

distance sublevel sets homology barcode

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The pipeline of topological data analysis

point cloud function topological spaces vector spaces

M ∶ R → Vect

intervals

distance sublevel sets homology barcode B(M)

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The pipeline of topological data analysis

point cloud function topological spaces vector spaces

M ∶ R → Vect

intervals

R → Mch

distance sublevel sets homology barcode B(M)

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The pipeline of topological data analysis

point cloud

P ⊂ Rd

function

f ∶ Rd → R

topological spaces

K ∶ R → Top

vector spaces

M ∶ R → Vect

intervals

R → Mch

distance x↦d(x,P) sublevel sets t↦f −1(−∞,t] homology t↦H∗(Kt;F) barcode B(M)

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Stability of persistence barcodes for functions

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

If two functions f , g ∶ K → R have distance ∥f − g∥∞ ≤ δ then there exists a δ-matching of their barcodes.

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Stability of persistence barcodes for functions

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

If two functions f , g ∶ K → R have distance ∥f − g∥∞ ≤ δ then there exists a δ-matching of their barcodes.

• matching A →

∣ B: bijection of subsets A′ ⊆ A, B′ ⊆ B

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Stability of persistence barcodes for functions

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

If two functions f , g ∶ K → R have distance ∥f − g∥∞ ≤ δ then there exists a δ-matching of their barcodes.

• matching A →

∣ B: bijection of subsets A′ ⊆ A, B′ ⊆ B

• δ-matching of barcodes:

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Stability of persistence barcodes for functions

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

If two functions f , g ∶ K → R have distance ∥f − g∥∞ ≤ δ then there exists a δ-matching of their barcodes.

δ

• matching A →

∣ B: bijection of subsets A′ ⊆ A, B′ ⊆ B

• δ-matching of barcodes:
• matched intervals have endpoints within distance ≤ δ

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Stability of persistence barcodes for functions

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

If two functions f , g ∶ K → R have distance ∥f − g∥∞ ≤ δ then there exists a δ-matching of their barcodes.

δ 2δ

• matching A →

∣ B: bijection of subsets A′ ⊆ A, B′ ⊆ B

• δ-matching of barcodes:
• matched intervals have endpoints within distance ≤ δ
• unmatched intervals have length ≤ 2δ

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Stability for functions in the big picture

Data

point cloud

Geometry

function

Topology

topological spaces

Algebra

vector spaces

Combinatorics

intervals

distance sublevel sets homology barcode

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Stability for functions in the big picture

Data

point cloud

Geometry

function

Topology

topological spaces

Algebra

vector spaces

Combinatorics

intervals

distance sublevel sets homology barcode

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Stability for functions in the big picture

Data

point cloud

Geometry

function

Topology

topological spaces

Algebra

vector spaces

Combinatorics

intervals

distance sublevel sets homology barcode

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Stability for functions in the big picture

Data

point cloud

Geometry

function

Topology

topological spaces

Algebra

vector spaces

Combinatorics

intervals

distance sublevel sets homology barcode

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Interleavings of sublevel sets

Let

• Ft = f −1(−∞, t],
• Gt = g−1(−∞, t].

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Interleavings of sublevel sets

Let

• Ft = f −1(−∞, t],
• Gt = g−1(−∞, t].

If ∥f − g∥∞ ≤ δ then Ft ⊆ Gt+δ and Gt ⊆ Ft+δ.

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Interleavings of sublevel sets

Let

• Ft = f −1(−∞, t],
• Gt = g−1(−∞, t].

If ∥f − g∥∞ ≤ δ then Ft ⊆ Gt+δ and Gt ⊆ Ft+δ. So the sublevel sets are δ-interleaved: Ft Ft+2δ Gt+δ Gt+3δ

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Interleavings of sublevel sets

Let

• Ft = f −1(−∞, t],
• Gt = g−1(−∞, t].

If ∥f − g∥∞ ≤ δ then Ft ⊆ Gt+δ and Gt ⊆ Ft+δ. So the sublevel sets are δ-interleaved: H∗(Ft) H∗(Ft+2δ) H∗(Gt+δ) H∗(Gt+3δ) Homology is a functor: homology groups are interleaved too.

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Persistence modules

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

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Persistence modules

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

• a vector space Mt for each t ∈ R

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Persistence modules

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

• a vector space Mt for each t ∈ R (in this talk: dimMt < ∞)

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Persistence modules

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

• a vector space Mt for each t ∈ R (in this talk: dimMt < ∞)
• a linear map Ms → Mt for each s ≤ t (transition maps)

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Persistence modules

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

• a vector space Mt for each t ∈ R (in this talk: dimMt < ∞)
• a linear map Ms → Mt for each s ≤ t (transition maps)
• respecting identity: (Mt → Mt) = idMt

and composition: (Ms → Mt) ○ (Mr → Ms) = (Mr → Mt)

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Persistence modules

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

• a vector space Mt for each t ∈ R (in this talk: dimMt < ∞)
• a linear map Ms → Mt for each s ≤ t (transition maps)
• respecting identity: (Mt → Mt) = idMt

and composition: (Ms → Mt) ○ (Mr → Ms) = (Mr → Mt) A morphism f ∶ M → N is a natural transformation:

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Persistence modules

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

• a vector space Mt for each t ∈ R (in this talk: dimMt < ∞)
• a linear map Ms → Mt for each s ≤ t (transition maps)
• respecting identity: (Mt → Mt) = idMt

and composition: (Ms → Mt) ○ (Mr → Ms) = (Mr → Mt) A morphism f ∶ M → N is a natural transformation:

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

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Persistence modules

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

• a vector space Mt for each t ∈ R (in this talk: dimMt < ∞)
• a linear map Ms → Mt for each s ≤ t (transition maps)
• respecting identity: (Mt → Mt) = idMt

and composition: (Ms → Mt) ○ (Mr → Ms) = (Mr → Mt) A morphism f ∶ M → N is a natural transformation:

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

Ms Mt Ns Nt

fs ft

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Interval Persistence Modules

Let K be a field. For an arbitrary interval I ⊆ R, define the interval persistence module C(I) by C(I)t = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ K if t ∈ I,

• therwise;

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Interval Persistence Modules

Let K be a field. For an arbitrary interval I ⊆ R, define the interval persistence module C(I) by C(I)t = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ K if t ∈ I,

• therwise;

C(I)s → C(I)t = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ idK if s, t ∈ I,

• therwise.

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The structure of persistence modules

Theorem (Crawley-Boewey 2012)

Let M be a persistence module with dimMt < ∞ for all t.

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The structure of persistence modules

Theorem (Crawley-Boewey 2012)

Let M be a persistence module with dimMt < ∞ for all t. Then M is interval-decomposable:

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The structure of persistence modules

Theorem (Crawley-Boewey 2012)

Let M be a persistence module with dimMt < ∞ for all t. Then M is interval-decomposable: there exists a unique collection of intervals B(M)

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The structure of persistence modules

Theorem (Crawley-Boewey 2012)

Let M be a persistence module with dimMt < ∞ for all t. Then M is interval-decomposable: there exists a unique collection of intervals B(M) such that M ≅ ⊕

I∈B(M)

C(I).

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The structure of persistence modules

Theorem (Crawley-Boewey 2012)

Let M be a persistence module with dimMt < ∞ for all t. Then M is interval-decomposable: there exists a unique collection of intervals B(M) such that M ≅ ⊕

I∈B(M)

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

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The structure of persistence modules

Theorem (Crawley-Boewey 2012)

Let M be a persistence module with dimMt < ∞ for all t. Then M is interval-decomposable: there exists a unique collection of intervals B(M) such that M ≅ ⊕

I∈B(M)

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

• Motivates use of homology with field coefficients

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Interleavings of persistence modules

Definition

Two persistence modules M and N are δ-interleaved

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Interleavings of persistence modules

Definition

Two persistence modules M and N are δ-interleaved if there are morphisms f ∶ M → N(δ), g ∶ N → M(δ)

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Interleavings of persistence modules

Definition

Two persistence modules M and N are δ-interleaved if there are morphisms f ∶ M → N(δ), g ∶ N → M(δ) such that this diagrams commutes for all t: Mt Mt+2δ Nt+δ Nt+3δ

ft ft+2δ gt+δ

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Interleavings of persistence modules

Definition

Two persistence modules M and N are δ-interleaved if there are morphisms f ∶ M → N(δ), g ∶ N → M(δ) such that this diagrams commutes for all t: Mt Mt+2δ Nt+δ Nt+3δ

ft ft+2δ gt+δ

• define M(δ) by M(δ)t = Mt+δ

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Interleavings of persistence modules

Definition

Two persistence modules M and N are δ-interleaved if there are morphisms f ∶ M → N(δ), g ∶ N → M(δ) such that this diagrams commutes for all t: Mt Mt+2δ Nt+δ Nt+3δ

ft ft+2δ gt+δ

• define M(δ) by M(δ)t = Mt+δ

(shift barcode to the left by δ)

B(M) B(M(δ))

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Algebraic stability of persistence barcodes

Theorem (Chazal et al. 2009, 2012)

If two persistence modules are δ-interleaved, then there exists a δ-matching of their barcodes.

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Algebraic stability of persistence barcodes

Theorem (Chazal et al. 2009, 2012)

If two persistence modules are δ-interleaved, then there exists a δ-matching of their barcodes.

δ 2δ

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Algebraic stability of persistence barcodes

Theorem (Chazal et al. 2009, 2012)

If two persistence modules are δ-interleaved, then there exists a δ-matching of their barcodes.

δ 2δ

• converse statement also holds (isometry theorem)

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Algebraic stability of persistence barcodes

Theorem (Chazal et al. 2009, 2012)

If two persistence modules are δ-interleaved, then there exists a δ-matching of their barcodes.

δ 2δ

• converse statement also holds (isometry theorem)
• indirect proof, 80 page paper (Chazal et al. 2012)

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Our approach

Our proof takes a different approach:

• direct proof (no interpolation, matching immediately

from interleaving)

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Our approach

Our proof takes a different approach:

• direct proof (no interpolation, matching immediately

from interleaving)

• shows how morphism induces a matching

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Our approach

Our proof takes a different approach:

• direct proof (no interpolation, matching immediately

from interleaving)

• shows how morphism induces a matching
• stability follows from properties of a single morphism,

not just from a pair of morphisms

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Our approach

Our proof takes a different approach:

• direct proof (no interpolation, matching immediately

from interleaving)

• shows how morphism induces a matching
• stability follows from properties of a single morphism,

not just from a pair of morphisms

• relies on partial functoriality of the induced matching

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The matching category

A matching σ ∶ S →

∣ T is a bijection S′ → T′, where S′ ⊆ S, T′ ⊆ T.

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The matching category

A matching σ ∶ S →

∣ T is a bijection S′ → T′, where S′ ⊆ S, T′ ⊆ T.

Composition of matchings σ ∶ S →

∣ T and τ ∶ T → ∣ U:

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The matching category

A matching 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 category Mch

• objects: sets
• morphisms: matchings

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Barcodes as matching diagrams

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

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Barcodes as matching diagrams

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

• For each real number t, let Bt be those intervals of B that

contain t, and

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Barcodes as matching diagrams

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

• For each real number t, let Bt be those intervals of B that

contain t, and

• for each s ≤ t, define the matching Bs →

∣ Bt

to be the identity on Bs ∩ Bt.

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Barcode matchings as natural transformations

We can regard certain matchings of barcodes σ ∶ A →

∣ B

as natural transformations of functors R → Mch.

• consider restrictions σt ∶ At →

∣ Bt of σ to At × Bt:

As At Bs Bt

σs σt

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Barcode matchings as natural transformations

We can regard certain matchings of barcodes σ ∶ A →

∣ B

as natural transformations of functors R → Mch.

• consider restrictions σt ∶ At →

∣ Bt of σ to At × Bt:

As At Bs Bt

σs σt

• requirement on the matching σ:

if I ∈ A is matched to J ∈ B, then I overlaps J to the right.

I J

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Barcode matchings as interleavings

We can regard a δ-matching of barcodes σ ∶ A →

∣ B

as a δ-interleaving of functors R → Mch: At At+2δ Bt+δ Bt+3δ

• each matching At →

∣ Bt+δ is the restriction of σ

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Stability via functoriality?

Ft Ft+2δ Gt+δ Gt+3δ

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Stability via functoriality?

H∗(Ft) H∗(Ft+2δ) H∗(Gt+δ) H∗(Gt+3δ)

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Stability via functoriality?

B(H∗(Ft)) B(H∗(Ft+2δ)) B(H∗(Gt+δ)) B(H∗(Gt+3δ))

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Stability via functoriality?

B(H∗(Ft)) B(H∗(Ft+2δ)) B(H∗(Gt+δ)) B(H∗(Gt+3δ))

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Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2014)

There exists no functor VectR → Mch sending each persistence module to its barcode.

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Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2014)

There exists no functor VectR → Mch sending each persistence module to its barcode.

Proposition

There exists no functor Vect → Mch sending each vector space of dimension d to a set of cardinality d.

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Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2014)

There exists no functor VectR → Mch sending each persistence module to its barcode.

Proposition

There exists no functor Vect → Mch sending each vector space of dimension d to a set of cardinality d.

• Such a functor would necessarily send a linear map of

rank r to a matching of cardinality r.

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Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2014)

There exists no functor VectR → Mch sending each persistence module to its barcode.

Proposition

There exists no functor Vect → Mch sending each vector space of dimension d to a set of cardinality d.

• Such a functor would necessarily send a linear map of

rank r to a matching of cardinality r.

• In particular, there is no natural choice of basis for vector

spaces

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Structure of submodules and quotient modules

Proposition (B, Lesnick 2013)

For a persistence submodule K ⊆ M:

• B(K) is obtained from B(M) by

moving left endpoints to the right,

B(M) B(K)

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Structure of submodules and quotient modules

Proposition (B, Lesnick 2013)

For a persistence submodule K ⊆ M:

• B(K) is obtained from B(M) by

moving left endpoints to the right,

B(M) B(K)

• B(M/K) is obtained from B(M) by

moving right endpoints to the left.

B(M) B(M/K)

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Structure of submodules and quotient modules

Proposition (B, Lesnick 2013)

For a persistence submodule K ⊆ M:

• B(K) is obtained from B(M) by

moving left endpoints to the right,

B(M) B(K)

• B(M/K) is obtained from B(M) by

moving right endpoints to the left.

B(M) B(M/K)

This yields canonical matchings between the barcodes: match bars with the same right endpoint (resp. left endpoint)

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Structure of submodules and quotient modules

Proposition (B, Lesnick 2013)

For a persistence submodule K ⊆ M:

• B(K) is obtained from B(M) by

moving left endpoints to the right,

B(M) B(K)

• B(M/K) is obtained from B(M) by

moving right endpoints to the left.

B(M) B(M/K)

This yields canonical matchings between the barcodes: match bars with the same right endpoint (resp. left endpoint)

• If multiple bars have same endpoint:

match in order of decreasing length

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Induced matchings

For any morphism f ∶ M → N between persistence modules:

• decompose into M ↠ imf ↪ N

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Induced matchings

For any morphism f ∶ M → N between persistence modules:

• decompose into M ↠ imf ↪ N
• imf ≅ M/kerf is a quotient of M

B(M) B(im f )

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Induced matchings

For any morphism f ∶ M → N between persistence modules:

• decompose into M ↠ imf ↪ N
• imf ≅ M/kerf is a quotient of M

B(im f ) B(N)

• imf is a submodule of N

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Induced matchings

For any morphism f ∶ M → N between persistence modules:

• decompose into M ↠ imf ↪ N
• imf ≅ M/kerf is a quotient of M

B(M) B(im f ) B(N)

• imf is a submodule of N
• Composing the canonical matchings yields

a matching B(f ) ∶ B(M) →

∣ B(N) induced by f

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Induced matchings

For any morphism f ∶ M → N between persistence modules:

• decompose into M ↠ imf ↪ N
• imf ≅ M/kerf is a quotient of M

B(M) B(im f ) B(N)

• imf is a submodule of N
• Composing the canonical matchings yields

a matching B(f ) ∶ B(M) →

∣ B(N) induced by f

This matching is functorial for injections: B(K ↪ M) = B(L ↪ M) ○ B(K ↪ L)

B(M) B(K) B(L)

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Induced matchings

For any morphism f ∶ M → N between persistence modules:

• decompose into M ↠ imf ↪ N
• imf ≅ M/kerf is a quotient of M

B(M) B(im f ) B(N)

• imf is a submodule of N
• Composing the canonical matchings yields

a matching B(f ) ∶ B(M) →

∣ B(N) induced by f

This matching is functorial for injections: B(K ↪ M) = B(L ↪ M) ○ B(K ↪ L)

B(M) B(K) B(L)

Similar for surjections.

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The induced matching theorem

Define Mє by shrinking bars of B(M) from the right by є.

є

B(M) B(Mє)

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The induced matching theorem

Define Mє by shrinking bars of B(M) from the right by є.

Lemma

Let f ∶ M → N be a morphism such that kerf is є-trivial (all bars of B(kerf ) are shorter than є). Then Mє is a quotient module of imf .

є

B(M) B(Mє) B(im f ) M Mє imf

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The induced matching theorem

Define єN by shrinking bars of B(N) from the left by є.

Lemma

Let f ∶ M → N be a morphism such that cokerf is є-trivial (all bars of B(cokerf ) are shorter than є). Then єN is a submodule of imf .

B(im f )

є

B(N) B(єN) N imf

єN

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The induced matching theorem

B(M) B(im f ) B(N) M N imf

f

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The induced matching theorem

є є

B(M) B(im f ) B(N) M N Mє imf

єN f

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The induced matching theorem

є є

B(M) B(N)

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The induced matching theorem

Theorem (B, Lesnick 2013)

Let f ∶ M → N be a morphism with kerf and cokerf є-trivial.

є є

B(M) B(N)

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The induced matching theorem

Theorem (B, Lesnick 2013)

Let f ∶ M → N be a morphism with kerf and cokerf є-trivial. Then each interval of length ≥ є is matched by B(f ).

є є

B(M) B(N)

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The induced matching theorem

Theorem (B, Lesnick 2013)

Let f ∶ M → N be a morphism with kerf and cokerf є-trivial. Then each interval of length ≥ є is matched by B(f ). If B(f ) matches [b, d) ∈ B(M) to [b′, d′) ∈ B(N), then b′ ≤ b ≤ b′ + є and d − є ≤ d′ ≤ d.

є є

B(M) B(N)

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The induced matching theorem

Let f ∶ M → N(δ) be an interleaving morphism. Then kerf and cokerf are 2δ-trivial.

2δ 2δ

B(M) B(N(δ))

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The induced matching theorem

Let f ∶ M → N(δ) be an interleaving morphism. Then kerf and cokerf are 2δ-trivial.

Corollary (Algebraic stability via induced matchings)

A δ-interleaving between persistence modules induces a δ-matching of their persistence barcodes.

B(M) B(N)

±δ ±δ

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Stability via induced matchings

B(M) B(N)

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Stability via induced matchings

B(M) B(N(δ))

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Stability via induced matchings

B(M) B(N(δ)) B(im f )

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Stability via induced matchings

B(M) B(N(δ)) B(im f )

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Stability via induced matchings

B(M) B(N(δ)) B(im f )

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Stability via induced matchings

B(M) B(N(δ))

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Stability via induced matchings

B(M) B(N)

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Stability via induced matchings

B(M) B(N)

Thanks for your attention!

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