Stability Properties
R
f
g
∞
a
b
s
s̄
X
2
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, and let ε = kf − gk∞ .
Gα := g −1 ((−∞, α])
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 ⊆ · · ·
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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
0
X
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 Dg f → Dg f 2ε .
2ε
⇒ d∞
(Dg
f,
Dg
f
) ≤ 2ε
B
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 Dg f → Dg f 2ε .
→ proof: show that the multiplicities of Dg f
and Dg 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)
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 (Dg f, Dg g) ≤ 8ε
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 (Dg f, Dg g) ≤ 8ε
Improvement:
(2n + 2)ε
(2n + 1)ε
2nε
(2n + 1)ε
(2nε − 2)
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 (Dg f, Dg g) ≤ 8ε
Improvement:
(2n + 2)ε
(2n + 1)ε
2nε
(2n + 1)ε
(2nε − 2)
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 (Dg f, Dg g) ≤ 8ε
Improvement:
(2n + 2)ε
(2n + 1)ε
2nε
(2n + 1)ε
(2nε − 2)
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 (Dg f, Dg g) ≤ 8ε
Improvement:
(2n + 2)ε
(2n + 1)ε
2nε
(2n + 1)ε
(2nε − 2)
3
Proof sketch
Fα := f −1 ((−∞, α])
Let f, g : X → R be q-tame, 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 (Dg f, Dg g) ≤ 8ε
Improvement:
d∞
B (Dg f, Dg g) ≤ 3ε
(2n + 2)ε
(2n + 1)ε
2nε
(2n + 1)ε
(2nε − 2)
3
Proof sketch
• 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
3