Connections
Morse theory
representation theory
discrete Morse theory
quiver theory
size theory
algebra
topology
topological persistence
linear systems
matrix multiplication
algorithms
1
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
dendrogram
scale
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
(single-linkage)
dendrogram is:
- informative
- unstable
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
dendrogram → barcode
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
dendrogram → barcode
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
dendrogram → barcode
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
barcode is:
- less (but still) informative
- more stable
0
2
4
6
8
10
12
14
16
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
barcode is:
- less (but still) informative
- more stable
- generalizable
0
4
8
12
16
20
24
28
32
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
barcode is:
- less (but still) informative
- more stable
- generalizable
0
4
8
12
16
20
24
28
32
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
barcode is:
- less (but still) informative
- more stable
- generalizable
0
4
8
12
16
20
24
28
32
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
barcode is:
- less (but still) informative
- more stable
- generalizable
0
4
8
12
16
20
24
28
32
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
barcode is:
- less (but still) informative
- more stable
- generalizable
0
4
8
12
16
20
24
28
32
2
∃ barcodes: intuition
(Agglomerative Hierarchical Clustering)
barcode is:
- less (but still) informative
- more stable
- generalizable
0
4
8
12
16
20
24
28
32
2
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
a
b
F1
a
b
a
b
a
b
a
b
a
b
d
c
d
c
d
c
d
c
d
c
F2
F3
F4
F5
F6
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
t
Ft := f −1 ((−∞, t])
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
Example 1: offsets filtration (nested family of unions of balls, cf. previous slide)
Example 2: simplicial filtration (nested family of simplicial complexes)
Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X → R)
R
f
R
3
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
topological level
(homology functor)
the functor is parametrized
by a field of coefficients, omitted in the notations
k
algebraic level
Persistence module: H∗ (F1 ) → H∗ (F2 ) → H∗ (F3 ) → H∗ (F4 ) → H∗ (F5 ) · · ·
4
∃ barcodes: decomposition theorems
F1I am
⊆ Findexing
⊆ family
F4 ⊆ over
F5 · ·the
· integers, but in fact it can be indexed ove
Note:Filtration:
for simplicity
2 ⊆ F3the
topological level
(homology functor)
the functor is parametrized
by a field of coefficients, omitted in the notations
k
algebraic level
Persistence module: H∗ (F1 ) → H∗ (F2 ) → H∗ (F3 ) → H∗ (F4 ) → H∗ (F5 ) · · ·
→ algebraic structure of module is described by a barcode
4
∃ barcodes: decomposition theorems
Example:
⊆
⊆
⊆
(1-homology functor)
k
k
( 10 )
/k
2 (0 1)
⊆
/k
( 01 )
/ k2
( 10 01 )
/ k2 · · ·
4
∃ barcodes: decomposition theorems
Example:
⊆
⊆
⊆
(1-homology functor)
k
k
( 10 )
/k
2 (0 1)
⊆
/k
( 01 )
/ k2
( 10 01 )
/ k2 · · ·
4
∃ barcodes: decomposition theorems
Theorem.
Let V be a persistence module over some index set T ⊆ R.
Then, V decomposes as a direct sum of interval modules IQ [b∗ , d∗ ]:
/0
0
{z
}
|
∗
d∗ ]
i>d∗the interval b
Note: the starsi<b
indicate
limits from below [bor∗ ,above,
depending on whether
0
/ ···
0
/0
}
0
/k
|
1
/ ···
{z
1
in the following cases:
• T is finite [Gabriel 1972] [Auslander 1974],
/k
}
0
V∼
=
/0
|
M
0
/ ···
{z
0
IQ [b∗j , d∗j ]
j∈J
• V that
is pointwise
space
Vt has
forward means
all arrowsfinite-dimensional
i → j satisfy i ≤ j,(every
with the
relation
i ≤ jfinite
≤ kĩ dimension)
≤k
[Webb 1985] [Crawley-Boevey 2012].
Moreover, when it exists, the decomposition is unique up to isomorphism and
permutation of the terms [Azumaya 1950].
(the barcode is a complete descriptor of the algebraic structure of V)
4
∃ barcodes: decomposition theorems
Theorem.
Let V be a persistence module over some index set T ⊆ R.
Then, V decomposes as a direct sum of interval modules IQ [b∗ , d∗ ]:
/0
0
{z
}
|
∗
d∗ ]
i>d∗the interval b
Note: the starsi<b
indicate
limits from below [bor∗ ,above,
depending on whether
0
/ ···
0
/0
}
0
/k
|
1
/ ···
{z
1
/k
}
0
/0
|
0
/ ···
{z
0
in the following cases:
• T is finite [Gabriel 1972] [Auslander 1974],
• V that
is pointwise
space
Vt has
forward means
all arrowsfinite-dimensional
i → j satisfy i ≤ j,(every
with the
relation
i ≤ jfinite
≤ kĩ dimension)
≤k
[Webb 1985] [Crawley-Boevey 2012].
Moreover, when it exists, the decomposition is unique up to isomorphism and
permutation of the terms [Azumaya 1950].
(the barcode is a complete descriptor of the algebraic structure of V)
4
∃ barcodes: decomposition theorems
Theorem.
Let V be a persistence module over some index set T ⊆ R.
Then, V decomposes as a direct sum of interval modules IQ [b∗ , d∗ ]:
/0
0
{z
}
|
∗
d∗ ]
i>d∗the interval b
Note: the starsi<b
indicate
limits from below [bor∗ ,above,
depending on whether
0
/ ···
0
/0
}
0
/k
|
1
/ ···
{z
1
/k
}
0
/0
|
0
/ ···
{z
0
in the following cases:
• T is finite [Gabriel 1972] [Auslander 1974],
• V that
is pointwise
space
Vt has
forward means
all arrowsfinite-dimensional
i → j satisfy i ≤ j,(every
with the
relation
i ≤ jfinite
≤ kĩ dimension)
≤k
[Webb 1985] [Crawley-Boevey 2012].
Moreover, when it exists, the decomposition is unique up to isomorphism and
permutation
the V
terms
[Azumaya
1950]. (finite T and finite-dim. V )
simple of
case:
is finitely
generated
t
(the barcode is a complete descriptor of the algebraic structure of V)
4
Computation of barcodes: matrix reduction
[Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005] . . .
3
Input: simplicial filtration
6
1
7
4
5
2
5
Computation of barcodes: matrix reduction
[Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005] . . .
3
Input: simplicial filtration
Output: boundary matrix
6
1
1
1
2
3
4
5
6
7
2
3
4
∗
∗
5
∗
∗
6
∗
7
4
5
2
7
∗
∗
∗
∗
5
Computation of barcodes: matrix reduction
[Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005] . . .
3
Input: simplicial filtration
Output: boundary matrix
reduced to column-echelon form
6
7
1
1
1
2
3
4
5
6
7
2
3
4
∗
∗
5
∗
∗
6
∗
7
∗
∗
∗
∗
1
1
2
3
4
5
6
7
5
4
2
3
4
∗
1
5
2
6
7
∗
1
∗
∗
1
5
Computation of barcodes: matrix reduction
[Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005] . . .
3
Input: simplicial filtration
Output: boundary matrix
reduced to column-echelon form
6
5
7
simplex pairs give finite intervals:
[2, 4), [3, 5), [6, 7)
1
4
2
unpaired simplices give infinite intervals: [1, +∞)
1
1
2
3
4
5
6
7
2
3
4
∗
∗
5
∗
∗
6
∗
7
∗
∗
∗
∗
1
1
2
3
4
5
6
7
2
3
4
∗
1
5
6
7
∗
1
∗
∗
1
5
Computation of barcodes: matrix reduction
[Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005] . . .
Input: simplicial filtration
Output: boundary matrix
reduced to column-echelon form
PLU factorization:
cubic worst-case time, near linear in practice because the boundary matrices tend to remain
• Gaussian elimination
adapted from [Bunch, Hopcroft 1974] → subcubic worst-case time, very complicated + very
• fast matrix multiplication (divide-and-conquer) [Bunch, Hopcroft 1974]
• random projections?
5
Computation of barcodes: matrix reduction
[Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005] . . .
Input: simplicial filtration
Output: boundary matrix
reduced to column-echelon form
PLU factorization:
cubic worst-case time, near linear in practice because the boundary matrices tend to remain
• Gaussian elimination
- PLEX / JavaPLEX (http://appliedtopology.github.io/javaplex/)
- Dionysus (http://www.mrzv.org/software/dionysus/)
- Perseus (http://www.sas.upenn.edu/~vnanda/perseus/)
- Gudhi (http://gudhi.gforge.inria.fr/)
- PHAT (https://bitbucket.org/phat- code/phat)
- DIPHA (https://github.com/DIPHA/dipha/)
- CTL (https://github.com/appliedtopology/ctl)
5
Stability of persistence barcodes
X topological
X → Rinfunction
the stability
property isspace,
easiest fto: describe
the functional setting
sublevel-sets filtration → barcode
barcode ≡ multiset of intervals
R
f
X
6
Stability of persistence barcodes
X topological
X → Rinfunction
the stability
property isspace,
easiest fto: describe
the functional setting
sublevel-sets filtration → barcode / diagram
barcode ≡ multiset of intervals
R
f
diagram ≡ multiset of points
∞
β
α
β
X
α
6
Stability of persistence barcodes
X topological space, f : X → R function
sublevel-sets filtration → barcode / diagram
R
f
g
∞
X
6
Stability of persistence barcodes
Theorem: For any pfd functions f, g : X → R,
d∞
b (dgm f, dgm g) ≤ kf − gk∞
R
f
g
∞
X
6
Stability of persistence barcodes
Theorem: For any pfd functions f, g : X → R,
d∞
b (dgm f, dgm g) ≤ kf − gk∞
Note: f, g do not have to have to be defined over the same domain X
but in the generalized case the statement is more subtle to state, so let me stick to this simpler version
R
f
g
∞
X
6
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Key observation: {Fα }α and {Gα }α are ε-interleaved w.r.t. inclusion:
∀α ∈ R, Gα−ε ⊆ Fα ⊆ Gα+ε
R
α+ε
α
X
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Key observation: {Fα }α and {Gα }α are ε-interleaved w.r.t. inclusion:
∀α ∈ R, Gα−ε ⊆ Fα ⊆ Gα+ε
R
→ Intuition: every homological feature that
appears/dies at time α in the filtration of f
appears/dies at time α + ε at the latest in the
filtration of g, and vice versa.
α+ε
α
X
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Key observation: {Fα }α and {Gα }α are ε-interleaved w.r.t. inclusion:
· · · ⊆ F0 ⊆ Gε ⊆ F2ε ⊆ · · · ⊆ G(2n−1)ε ⊆ F2nε ⊆ G(2n+1)ε ⊆ · · ·
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Key observation: {Fα }α and {Gα }α are ε-interleaved w.r.t. inclusion:
· · · ⊆ F0 ⊆ Gε ⊆ F2ε ⊆ · · · ⊆ G(2n−1)ε ⊆ F2nε ⊆ G(2n+1)ε ⊆ · · ·
- the filtration {F
}
is a 2ε-discretization of {F }
i.e. it is a sequence of snapshots of2nε
the sub-level
of 2ε
α α∈R
n∈Z set filtration of f , taken preiodically at a period
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Key observation: {Fα }α and {Gα }α are ε-interleaved w.r.t. inclusion:
· · · ⊆ F0 ⊆ Gε ⊆ F2ε ⊆ · · · ⊆ G(2n−1)ε ⊆ F2nε ⊆ G(2n+1)ε ⊆ · · ·
- the filtration {F
}
is a 2ε-discretization of {F }
i.e. it is a sequence of snapshots of2nε
the sub-level
of 2ε
α α∈R
n∈Z set filtration of f , taken preiodically at a period
- the filtration {G(2n+1)ε }n∈Z is a 2ε-discretization of {Gα }α∈R
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Key observation: {Fα }α and {Gα }α are ε-interleaved w.r.t. inclusion:
· · · ⊆ F0 ⊆ Gε ⊆ F2ε ⊆ · · · ⊆ G(2n−1)ε ⊆ F2nε ⊆ G(2n+1)ε ⊆ · · ·
- the filtration {F
}
is a 2ε-discretization of {F }
i.e. it is a sequence of snapshots of2nε
the sub-level
of 2ε
α α∈R
n∈Z set filtration of f , taken preiodically at a period
- the filtration {G(2n+1)ε }n∈Z is a 2ε-discretization of {Gα }α∈R
- both filtrations are 2ε-discretizations of {Hnε }n∈Z ,where Hnε =
Fnε if n is even
Gnε if n is odd
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Key observation: {Fα }α and {Gα }α are ε-interleaved w.r.t. inclusion:
· · · ⊆ F0 ⊆ Gε ⊆ F2ε ⊆ · · · ⊆ G(2n−1)ε ⊆ F2nε ⊆ G(2n+1)ε ⊆ · · ·
- the filtration {F
}
is a 2ε-discretization of {F }
i.e. it is a sequence of snapshots of2nε
the sub-level
of 2ε
α α∈R
n∈Z set filtration of f , taken preiodically at a period
- the filtration {G(2n+1)ε }n∈Z is a 2ε-discretization of {Gα }α∈R
- both filtrations are 2ε-discretizations of {Hnε }n∈Z ,where Hnε =
Fnε if n is even
Gnε if n is odd
→ goal: bound distances between diagrams of filtrations and discretizations
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Discretization ⇒ pixelization effect on the barcodes / diagrams:
R
18ε
16ε
0
2ε 4ε
8ε
16ε
14ε
12ε
10ε
8ε
6ε
4ε
2ε
f 2ε
f
0
X
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Discretization ⇒ pixelization effect on the barcodes / diagrams:
Pixelization map: ∀α ≤ β,
π2ε (α, β) =
0
2ε 4ε
8ε
16ε

β
β
α
α
(d
e2ε,
d
e2ε)
if
d
e
>
d
e

2ε
2ε
2ε
2ε

( α+β
,
2
α+β
)
2
β
α
if d 2ε
e = d 2ε
e
Theorem: If f : X → R is q-tame, then
π2ε induces a bijection dgm f → dgm f 2ε .
2ε
⇒ d∞
(dgm
f,
dgm
f
) ≤ 2ε
b
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Discretization ⇒ pixelization effect on the barcodes / diagrams:
Pixelization map: ∀α ≤ β,
π2ε (α, β) =
0
2ε 4ε
8ε
16ε

β
β
α
α
(d
e2ε,
d
e2ε)
if
d
e
>
d
e

2ε
2ε
2ε
2ε

( α+β
,
2
α+β
)
2
β
α
if d 2ε
e = d 2ε
e
Theorem: If f : X → R is q-tame, then
π2ε induces a bijection dgm f → dgm f 2ε .
→ proof: show that the multiplicities of dgm f
and dgm f 2ε are the same inside each grid cell
that does not intersect the diagonal.
(3)
(1)
The case of diagonal cells is trivial.
(1)
(1)
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Back to interleaved filtrations:
· · · ⊆ F0 ⊆ Gε ⊆ F2ε ⊆ · · · ⊆ G(2n−1)ε ⊆ F2nε ⊆ G(2n+1)ε ⊆ · · ·
Hnε =
Fnε if n is even
Gnε if n is odd
Previous theorem + triangle inequality ⇒ d∞
b (dgm f, dgm g) ≤ 8ε
7
Intuition behind the proof
Fα := f −1 ((−∞, α])
Let f, g : X → R be pfd, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
• Back to interleaved filtrations:
· · · ⊆ F0 ⊆ Gε ⊆ F2ε ⊆ · · · ⊆ G(2n−1)ε ⊆ F2nε ⊆ G(2n+1)ε ⊆ · · ·
Hnε =
Fnε if n is even
Gnε if n is odd
Previous theorem + triangle inequality ⇒ d∞
b (dgm f, dgm g) ≤ 8ε
Improvement:
d∞
b (dgm f, dgm g) ≤ 3ε
(2n + 2)ε
(2n + 1)ε
2nε
(2n + 1)ε
(2nε − 2)
7
Intuition behind the proof
• Comments:
I chose to present this one because it is the most geometrically flavored, and the one requiring the least amount of extra background material
- sketch of proof based on [Chazal, Cohen-Steiner, Glisse, Guibas, O. 2009].
- uses only the fact that F, G are interleaved over the scale εZ.
this is called weakly ε-interleaved
- bound 3ε is tight under this assumption.
−3ε
−2ε
−ε
0
ε
2ε
3ε
δ
δ
u
Here Fis a counter-example.
I won’t comment on it due to the lack of time.
v
G
u
v
7
Intuition behind the proof
• Comments:
I chose to present this one because it is the most geometrically flavored, and the one requiring the least amount of extra background material
- sketch of proof based on [Chazal, Cohen-Steiner, Glisse, Guibas, O. 2009].
- uses only the fact that F, G are interleaved over the scale εZ.
this is called weakly ε-interleaved
- bound 3ε is tight under this assumption.
∀α ∈ R, Gα−ε ⊆ Fα ⊆ Gα+ε
- full interleaving hypothesis gives bound ε via an interpolation argument:
Here is a counter-example. I won’t comment on it due to the lack of time.
# at the functional level (requires extra conditions)
[Cohen-Steiner, Edelsbrunner, Harer 2005]
# at the algebraic level directly (→ isometry theorem)
[Chazal, Cohen-Steiner, Glisse, Guibas, O. 2009]
[Chazal, de Silva, Glisse, O. 2016]
[Bauer, Lesnick 2015]
[Botnan, Lesnick 2016]
...
7
Recap’
3 pillars to the theory (topological persistence):
• decomposition theorems (∃ barcodes)
• algorithms (computation of barcodes)
• stability theorems (barcodes as stable descriptors)
8