Topological Data Analysis
Part II: Stability of persistence barcodes
Ulrich Bauer
TUM
February 5, 2015
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What is persistent homology?
0.1
0.2
0.4
0.8
δ
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What is persistent homology?
0.1
0.2
0.4
0.8
δ
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What is persistent homology?
0.1
0.2
0.4
0.8
δ
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What is persistent homology?
0.1
0.2
0.4
0.8
δ
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What is persistent homology?
0.1
0.2
0.4
0.8
δ
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What is persistent homology?
0.1
0.2
0.4
0.8
δ
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The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
homology
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
P ⊂ Rd
distance
function
sublevel sets
topological spaces
homology
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
distance
P ⊂ Rd
x↦d(x,P)
function
sublevel sets
topological spaces
homology
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
distance
function
P ⊂ Rd
x↦d(x,P)
f ∶ Rd → R
sublevel sets
topological spaces
homology
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
distance
function
f ∶ Rd → R
sublevel sets
topological spaces
homology
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
f ∶ Rd → R
t↦f −1 (−∞,t]
topological spaces
homology
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
f ∶ Rd → R
t↦f −1 (−∞,t]
K ∶ R → Top
homology
vector spaces
barcode
intervals
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The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
K ∶ R → Top
homology
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
homology
K ∶ R → Top
t↦H∗ (Kt ;F)
vector spaces
barcode
intervals
4 / 31
The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
homology
vector spaces
K ∶ R → Top
t↦H∗ (Kt ;F)
M ∶ R → Vect
barcode
intervals
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The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
homology
vector spaces
M ∶ R → Vect
barcode
intervals
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The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
homology
vector spaces
barcode
M ∶ R → Vect
B(M)
intervals
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The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
homology
vector spaces
barcode
intervals
M ∶ R → Vect
B(M)
R → Mch
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The pipeline of topological data analysis
point cloud
distance
function
sublevel sets
topological spaces
homology
vector spaces
barcode
intervals
P ⊂ Rd
x↦d(x,P)
f ∶ Rd → R
t↦f −1 (−∞,t]
K ∶ R → Top
t↦H∗ (Kt ;F)
M ∶ R → Vect
B(M)
R → Mch
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Stability of persistence
barcodes
5 / 31
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.
6 / 31
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
∣
6 / 31
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:
6 / 31
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
6 / 31
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.
2δ
δ
• 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
distance
Geometry
function
sublevel sets
Topology
topological spaces
homology
Algebra
vector spaces
barcode
Combinatorics
intervals
7 / 31
Stability for functions in the big picture
Data
point cloud
distance
Geometry
function
sublevel sets
Topology
topological spaces
homology
Algebra
vector spaces
barcode
Combinatorics
intervals
7 / 31
Stability for functions in the big picture
Data
point cloud
distance
Geometry
function
sublevel sets
Topology
topological spaces
homology
Algebra
vector spaces
barcode
Combinatorics
intervals
7 / 31
Stability for functions in the big picture
Data
point cloud
distance
Geometry
function
sublevel sets
Topology
topological spaces
homology
Algebra
vector spaces
barcode
Combinatorics
intervals
7 / 31
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: dim Mt < ∞)
9 / 31
Persistence modules
A persistence module M is a diagram (functor) R → Vect:
• a vector space Mt for each t ∈ R (in this talk: dim Mt < ∞)
• a linear map Ms → Mt for each s ≤ t (transition maps)
9 / 31
Persistence modules
A persistence module M is a diagram (functor) R → Vect:
• a vector space Mt for each t ∈ R (in this talk: dim Mt < ∞)
• a linear map Ms → Mt for each s ≤ t (transition maps)
• respecting identity: (Mt → Mt ) = idMt
and composition: (Ms → Mt ) ○ (Mr → Ms ) = (Mr → Mt )
9 / 31
Persistence modules
A persistence module M is a diagram (functor) R → Vect:
• a vector space Mt for each t ∈ R (in this talk: dim Mt < ∞)
• 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:
9 / 31
Persistence modules
A persistence module M is a diagram (functor) R → Vect:
• a vector space Mt for each t ∈ R (in this talk: dim Mt < ∞)
• 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: dim Mt < ∞)
• 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
fs
Ns
Mt
ft
Nt
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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
⎧
⎪
⎪K
C(I)t = ⎨
⎪
⎪
⎩0
if t ∈ I,
otherwise;
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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
⎧
⎪
⎪K if t ∈ I,
C(I)t = ⎨
⎪
⎪
⎩0 otherwise;
⎧
⎪
⎪idK if s, t ∈ I,
C(I)s → C(I)t = ⎨
⎪
⎪
otherwise.
⎩0
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The structure of persistence modules
Theorem (Crawley-Boewey 2012)
Let M be a persistence module with dim Mt < ∞ for all t.
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The structure of persistence modules
Theorem (Crawley-Boewey 2012)
Let M be a persistence module with dim Mt < ∞ 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 dim Mt < ∞ 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 dim Mt < ∞ for all t.
Then M is interval-decomposable:
there exists a unique collection of intervals B(M) such that
M ≅ ⊕ C(I).
I∈B(M)
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The structure of persistence modules
Theorem (Crawley-Boewey 2012)
Let M be a persistence module with dim Mt < ∞ for all t.
Then M is interval-decomposable:
there exists a unique collection of intervals B(M) such that
M ≅ ⊕ C(I).
I∈B(M)
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 dim Mt < ∞ for all t.
Then M is interval-decomposable:
there exists a unique collection of intervals B(M) such that
M ≅ ⊕ C(I).
I∈B(M)
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)
12 / 31
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
ft
ft+2δ
gt+δ
Nt+δ
Nt+3δ
12 / 31
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
ft
ft+2δ
gt+δ
Nt+δ
Nt+3δ
• define M(δ) by M(δ)t = Mt+δ
12 / 31
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
ft
ft+2δ
gt+δ
Nt+δ
Nt+3δ
B(M)
B(M(δ))
• define M(δ) by M(δ)t = Mt+δ
(shift barcode to the left by δ)
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.
2δ
δ
13 / 31
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.
2δ
δ
• converse statement also holds (isometry theorem)
13 / 31
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.
2δ
δ
• converse statement also holds (isometry theorem)
• indirect proof, 80 page paper (Chazal et al. 2012)
13 / 31
Our approach
Our proof takes a different approach:
• direct proof (no interpolation, matching immediately
from interleaving)
14 / 31
Our approach
Our proof takes a different approach:
• direct proof (no interpolation, matching immediately
from interleaving)
• shows how morphism induces a matching
14 / 31
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
14 / 31
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
14 / 31
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 D as a functor R → Mch:
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0.8
δ
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Barcodes as matching diagrams
We can regard a barcode D as a functor R → Mch:
• For each real number t, let Dt be those intervals of D that
contain t, and
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0.2
0.4
0.8
δ
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Barcodes as matching diagrams
We can regard a barcode D as a functor R → Mch:
• For each real number t, let Dt be those intervals of D that
contain t, and
• for each s ≤ t, define the matching Ds → Dt
∣
to be the identity on Ds ∩ Dt .
0.1
0.2
0.4
0.8
δ
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Barcode matchings as interleavings
We can regard matchings of barcodes σ ∶ C → D as natural
transformations.
• consider restrictions σt ∶ Ct → Dt of σ to Ct × Dt :
∣
∣
Cs
σs
Ds
Ct
σt
Dt
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Barcode matchings as interleavings
We can regard matchings of barcodes σ ∶ C → D as natural
transformations.
• consider restrictions σt ∶ Ct → Dt of σ to Ct × Dt :
∣
∣
Cs
Ct
σs
σt
Ds
Dt
This way, we can regard a δ-matching of barcodes C → D
as a δ-interleaving of functors R → Mch:
∣
Ct
Ct+2δ
Dt+δ
Dt+3δ
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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 )
18 / 31
Stability via functoriality?
B(H∗ (Ft ))
B(H∗ (Ft+2δ ))
B(H∗ (Gt+δ ))
B(H∗ (Gt+3δ ))
18 / 31
Stability via functoriality?
B(H∗ (Ft ))
B(H∗ (Ft+2δ ))
B(H∗ (Gt+δ ))
B(H∗ (Gt+3δ ))
18 / 31
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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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)
20 / 31
Structure of submodules and quotient modules
Proposition (B, Lesnick 2013)
For a persistence submodule K ⊆ M:
• B(K) is obtained from B(M) by
B(M)
B(K)
moving left endpoints to the right,
• B(M/K) is obtained from B(M) by
moving right endpoints to the left.
B(M)
B(M/K)
20 / 31
Structure of submodules and quotient modules
Proposition (B, Lesnick 2013)
For a persistence submodule K ⊆ M:
• B(K) is obtained from B(M) by
B(M)
B(K)
moving left endpoints to the right,
• 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
B(M)
B(K)
moving left endpoints to the right,
• 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
20 / 31
Induced matchings
For any morphism f ∶ M → N between persistence modules:
• decompose into M ↠ im f ↪ N
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Induced matchings
For any morphism f ∶ M → N between persistence modules:
• decompose into M ↠ im f ↪ N
• im f ≅ M/ ker f is a quotient of M
B(M)
B(im f )
21 / 31
Induced matchings
For any morphism f ∶ M → N between persistence modules:
• decompose into M ↠ im f ↪ N
• im f ≅ M/ ker f is a quotient of M
• im f is a submodule of N
B(im f )
B(N)
21 / 31
Induced matchings
For any morphism f ∶ M → N between persistence modules:
• decompose into M ↠ im f ↪ N
• im f ≅ M/ ker f is a quotient of M
• im f is a submodule of N
B(M)
B(im f )
B(N)
• Composing the canonical matchings yields
a matching B(f ) ∶ B(M) → B(N) induced by f
∣
21 / 31
Induced matchings
For any morphism f ∶ M → N between persistence modules:
• decompose into M ↠ im f ↪ N
• im f ≅ M/ ker f is a quotient of M
• im f is a submodule of N
B(M)
B(im f )
B(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(L)
B(K)
21 / 31
Induced matchings
For any morphism f ∶ M → N between persistence modules:
• decompose into M ↠ im f ↪ N
• im f ≅ M/ ker f is a quotient of M
• im f is a submodule of N
B(M)
B(im f )
B(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(L)
B(K)
Similar for surjections.
21 / 31
The induced matching theorem
Define Mє by shrinking bars of B(M) from the right by є.
Say K is є-trivial if all bars of B(K) are shorter than є.
B(M)
є
B(M є)
22 / 31
The induced matching theorem
Define Mє by shrinking bars of B(M) from the right by є.
Say K is є-trivial if all bars of B(K) are shorter than є.
Lemma
Let f ∶ M → N be a morphism such that ker f is є-trivial.
Then Mє is a quotient module of im f .
B(M)
є
B(im f )
B(M є)
M
Mє
im f
22 / 31
The induced matching theorem
Define єN by shrinking bars of B(N) from the left by є.
Say K is є-trivial if all bars of B(K) are shorter than є.
Lemma
Let f ∶ M → N be a morphism such that coker f is є-trivial.
Then єN is a submodule of im f .
є
B(єN)
B(im f )
B(N)
N
im f
є
N
22 / 31
The induced matching theorem
B(M)
B(im f )
B(N)
M
f
N
im f
22 / 31
The induced matching theorem
є
B(M)
B(im f )
B(N)
є
M
Mє
f
im f
N
є
N
22 / 31
The induced matching theorem
є
B(M)
є
B(N)
22 / 31
The induced matching theorem
Theorem (B, Lesnick 2013)
Let f ∶ M → N be a morphism with ker f and coker f є-trivial.
є
B(M)
є
B(N)
22 / 31
The induced matching theorem
Theorem (B, Lesnick 2013)
Let f ∶ M → N be a morphism with ker f and coker f є-trivial.
Then each interval of length ≥ є is matched by B(f ).
є
B(M)
є
B(N)
22 / 31
The induced matching theorem
Theorem (B, Lesnick 2013)
Let f ∶ M → N be a morphism with ker f and coker f 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
B(M)
є
B(N)
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The induced matching theorem
Let f ∶ M → N(δ) be an interleaving morphism.
Then ker f and coker f are 2δ-trivial.
2δ
B(M)
2δ
B(N(δ))
22 / 31
The induced matching theorem
Let f ∶ M → N(δ) be an interleaving morphism.
Then ker f and coker f are 2δ-trivial.
Corollary (Algebraic stability via induced matchings)
A δ-interleaving between persistence modules induces
a δ-matching of their persistence barcodes.
±δ
B(M)
±δ
B(N)
22 / 31
Stability via induced matchings
B(M)
B(N)
23 / 31
Stability via induced matchings
B(M)
B(N(δ))
23 / 31
Stability via induced matchings
B(M)
B(im f )
B(N(δ))
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Stability via induced matchings
B(M)
B(im f )
B(N(δ))
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Stability via induced matchings
B(M)
B(im f )
B(N(δ))
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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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Simplification of functions
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Topological simplification of functions
Consider the following problem:
Problem (Topological simplification)
Given a function f and a real number δ ≥ 0, find a function fδ
subject to ∥fδ − f ∥∞ ≤ δ with minimal number of critical points.
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Topological simplification of functions
Consider the following problem:
Problem (Topological simplification)
Given a function f and a real number δ ≥ 0, find a function fδ
subject to ∥fδ − f ∥∞ ≤ δ with minimal number of critical points.
(Discrete) Morse theory:
• Relates critical points to homology of sublevel set
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Topological simplification of functions
Consider the following problem:
Problem (Topological simplification)
Given a function f and a real number δ ≥ 0, find a function fδ
subject to ∥fδ − f ∥∞ ≤ δ with minimal number of critical points.
(Discrete) Morse theory:
• Relates critical points to homology of sublevel set
• Provides method for canceling pairs of critical points
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Topological simplification of functions
Consider the following problem:
Problem (Topological simplification)
Given a function f and a real number δ ≥ 0, find a function fδ
subject to ∥fδ − f ∥∞ ≤ δ with minimal number of critical points.
(Discrete) Morse theory:
• Relates critical points to homology of sublevel set
• Provides method for canceling pairs of critical points
Persistent homology:
• Relates homology of different sublevel set
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Topological simplification of functions
Consider the following problem:
Problem (Topological simplification)
Given a function f and a real number δ ≥ 0, find a function fδ
subject to ∥fδ − f ∥∞ ≤ δ with minimal number of critical points.
(Discrete) Morse theory:
• Relates critical points to homology of sublevel set
• Provides method for canceling pairs of critical points
Persistent homology:
• Relates homology of different sublevel set
• Identifies pairs of critical points (birth and death of
homological feature)
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Persistence and discrete Morse theory
By stability of persistence barcodes:
Proposition
The critical points with persistence > 2δ provide a lower bound on
the number of critical points of any function g with ∥g − f ∥∞ ≤ δ.
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Persistence and discrete Morse theory
By stability of persistence barcodes:
Proposition
The critical points with persistence > 2δ provide a lower bound on
the number of critical points of any function g with ∥g − f ∥∞ ≤ δ.
Theorem (B., Lange, Wardetzky, 2011)
Let f be a function on a surface and let δ > 0.
Construct a function fδ from f by canceling all persistence pairs
with persistence ≤ 2δ.
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Persistence and discrete Morse theory
By stability of persistence barcodes:
Proposition
The critical points with persistence > 2δ provide a lower bound on
the number of critical points of any function g with ∥g − f ∥∞ ≤ δ.
Theorem (B., Lange, Wardetzky, 2011)
Let f be a function on a surface and let δ > 0.
Construct a function fδ from f by canceling all persistence pairs
with persistence ≤ 2δ. Then fδ satisfies
∥fδ − f ∥∞ ≤ δ
and achieves the lower bound on the number of critical points.
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Persistence and discrete Morse theory
By stability of persistence barcodes:
Proposition
The critical points with persistence > 2δ provide a lower bound on
the number of critical points of any function g with ∥g − f ∥∞ ≤ δ.
Theorem (B., Lange, Wardetzky, 2011)
Let f be a function on a surface and let δ > 0.
Construct a function fδ from f by canceling all persistence pairs
with persistence ≤ 2δ. Then fδ satisfies
∥fδ − f ∥∞ ≤ δ
and achieves the lower bound on the number of critical points.
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Persistence and discrete Morse theory
By stability of persistence barcodes:
Proposition
The critical points with persistence > 2δ provide a lower bound on
the number of critical points of any function g with ∥g − f ∥∞ ≤ δ.
Theorem (B., Lange, Wardetzky, 2011)
Let f be a function on a surface and let δ > 0.
Construct a function fδ from f by canceling all persistence pairs
with persistence ≤ 2δ. Then fδ satisfies
∥fδ − f ∥∞ ≤ δ
and achieves the lower bound on the number of critical points.
• Does not hold for general complexes
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Geometric stability
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Čech and Vietoris–Rips complexes
Let X ⊂ Rd . Given δ > 0, consider
Cechδ (X) = {Q ⊆ X ∣ ⋂ Bδ (p) ≠ ∅}
p∈Q
Ripsδ (X) = {Q ⊆ X ∶ x, y ∈ Q ⇒ dM (x, y) ≤ δ}
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Čech and Vietoris–Rips complexes
Let X ⊂ Rd . Given δ > 0, consider
Cechδ (X) = {Q ⊆ X ∣ ⋂ Bδ (p) ≠ ∅}
p∈Q
Ripsδ (X) = {Q ⊆ X ∶ x, y ∈ Q ⇒ dM (x, y) ≤ δ}
Compare:
• Čech complex: thresholded by circumradius
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Čech and Vietoris–Rips complexes
Let X ⊂ Rd . Given δ > 0, consider
Cechδ (X) = {Q ⊆ X ∣ ⋂ Bδ (p) ≠ ∅}
p∈Q
Ripsδ (X) = {Q ⊆ X ∶ x, y ∈ Q ⇒ dM (x, y) ≤ δ}
Compare:
• Čech complex: thresholded by circumradius
• Vietoris–Rips complex: thresholded by diameter
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Čech and Vietoris–Rips complexes
Let X ⊂ Rd . Given δ > 0, consider
Cechδ (X) = {Q ⊆ X ∣ ⋂ Bδ (p) ≠ ∅}
p∈Q
Ripsδ (X) = {Q ⊆ X ∶ x, y ∈ Q ⇒ dM (x, y) ≤ δ}
Compare:
• Čech complex: thresholded by circumradius
• Vietoris–Rips complex: thresholded by diameter
We have:
Cech δ ⊆ Ripsδ ⊆ Cechϑd δ
2
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Čech and Vietoris–Rips complexes
Let X ⊂ Rd . Given δ > 0, consider
Cechδ (X) = {Q ⊆ X ∣ ⋂ Bδ (p) ≠ ∅}
p∈Q
Ripsδ (X) = {Q ⊆ X ∶ x, y ∈ Q ⇒ dM (x, y) ≤ δ}
Compare:
• Čech complex: thresholded by circumradius
• Vietoris–Rips complex: thresholded by diameter
We have:
where ϑd =
√
Cech δ ⊆ Ripsδ ⊆ Cechϑd δ
2
d
2d+2
is the largest circumradius of any simplex
with diameter 1.
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Čech and Vietoris–Rips complexes
Let X ⊂ Rd . Given δ > 0, consider
Cechδ (X) = {Q ⊆ X ∣ ⋂ Bδ (p) ≠ ∅}
p∈Q
Ripsδ (X) = {Q ⊆ X ∶ x, y ∈ Q ⇒ dM (x, y) ≤ δ}
Compare:
• Čech complex: thresholded by circumradius
• Vietoris–Rips complex: thresholded by diameter
We have:
where ϑd =
√
Cech δ ⊆ Ripsδ ⊆ Cechϑd δ
2
d
2d+2
is the largest circumradius of any simplex
with diameter 1. Note that 21 ≤ ϑd ≤
√
2
2 .
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Interleaving of Čech and Rips complexes
Using the logarithmic scale t = log δ:
Cechexp t ⊆ Rips2 exp(t+log ϑd ) ⊆ Cechexp(t+2 log ϑd )
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Interleaving of Čech and Rips complexes
Using the logarithmic scale t = log δ:
Cechexp t ⊆ Rips2 exp(t+log ϑd ) ⊆ Cechexp(t+2 log ϑd )
Corollary
The persistence modules H∗ (Cechexp t (P)) and H∗ (Rips2 exp t (P))
are (log ϑd )-interleaved.
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Stability of Čech and Rips persistence
Proposition
Let P, Ω ⊂ Rn . Assume that P and Ω have Hausdorff distance < δ
(i.e., P ⊆ Bδ (Ω) and Ω ⊆ Bδ (P)).
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Stability of Čech and Rips persistence
Proposition
Let P, Ω ⊂ Rn . Assume that P and Ω have Hausdorff distance < δ
(i.e., P ⊆ Bδ (Ω) and Ω ⊆ Bδ (P)).
Then P and Ω have δ-close Čech persistence, i.e.,
H∗ (Cecht (Ω)) and H∗ (Cecht (P)) are δ-interleaved.
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Stability of Čech and Rips persistence
Proposition
Let P, Ω ⊂ Rn . Assume that P and Ω have Hausdorff distance < δ
(i.e., P ⊆ Bδ (Ω) and Ω ⊆ Bδ (P)).
Then P and Ω have δ-close Čech persistence, i.e.,
H∗ (Cecht (Ω)) and H∗ (Cecht (P)) are δ-interleaved.
Theorem (Chazal, de Silva, Oudot 2013)
Let P, Ω be metric spaces. Assume that P and Ω have
Gromov-Hausdorff distance < 2δ
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Stability of Čech and Rips persistence
Proposition
Let P, Ω ⊂ Rn . Assume that P and Ω have Hausdorff distance < δ
(i.e., P ⊆ Bδ (Ω) and Ω ⊆ Bδ (P)).
Then P and Ω have δ-close Čech persistence, i.e.,
H∗ (Cecht (Ω)) and H∗ (Cecht (P)) are δ-interleaved.
Theorem (Chazal, de Silva, Oudot 2013)
Let P, Ω be metric spaces. Assume that P and Ω have
Gromov-Hausdorff distance < 2δ (i.e., P ⊆ Bδ (Ω) and Ω ⊆ Bδ (P)
for some isometric embeddings of both P, Ω into some metric
space Z).
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Stability of Čech and Rips persistence
Proposition
Let P, Ω ⊂ Rn . Assume that P and Ω have Hausdorff distance < δ
(i.e., P ⊆ Bδ (Ω) and Ω ⊆ Bδ (P)).
Then P and Ω have δ-close Čech persistence, i.e.,
H∗ (Cecht (Ω)) and H∗ (Cecht (P)) are δ-interleaved.
Theorem (Chazal, de Silva, Oudot 2013)
Let P, Ω be metric spaces. Assume that P and Ω have
Gromov-Hausdorff distance < 2δ (i.e., P ⊆ Bδ (Ω) and Ω ⊆ Bδ (P)
for some isometric embeddings of both P, Ω into some metric
space Z).
Then P and Ω have δ-close Rips persistence.
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