Dagstuhl - March 20, 2012
Applications of Combinatorial Topology to Computer Science
Persistence based signatures for
compact metric spaces
Frédéric Chazal
Joint work with Vin de Silva and Steve Oudot
Introduction
Problem: how to compare topological properties of close metric
spaces (X, dX ) and (Y, dY )?
Introduction
Problem: how to compare topological properties of close metric
spaces (X, dX ) and (Y, dY )?
• Gromov-Hausdorff distance between (X, dX ) and (Y, dY )
For ε ≥ 0, C ⊂ X × Y is an ε-correspondence if
i) ∀x ∈ X, ∃y ∈ Y s. t. (x, y) ∈ C;
ii) ∀y ∈ Y , ∃x ∈ X s. t. (x, y) ∈ C;
iii) ∀(x, y), (x0 , y 0 ) ∈ C, |dX (x, x0 ) − dY (y, y 0 )| ≤ ε.
1
dGH (X, Y ) = inf{ε ≥ 0 : there exists an ε-correspondence between Xand Y }
2
Introduction
∞
0.75
0.5
0.25
0
0
0-dimensional homology generators
1-dimensional homology generators
• Scale dependent approach considering the Vietoris-Rips complexes built on X
and Y .
For a ∈ R, Rips(X, a) is the simplicial complex with vertex set X
defined by
[x0 , x1 , · · · , xk ] ∈ Rips(X, a) ⇔ dX (xi , xj ) ≤ a, for all i, j
• Compare the persistence of the filtrations HRips(X) and HRips(Y ) where
H = Hp (−; F) is the simplicial homology functor in dimension p over a field
F.
Previous works
• J.-C. Hausmann 1995
• J. Latschev 2001
• D. Attali, A. Lieutier 2011
When (X, dX ) is a riemannian manifold
(or a sufficiently regular subset of Rd ),
Rips(X, a) is homotopy equivalent to X
for small enough a > 0.
• F. C., D. Cohen-Steiner, L. Guibas, F.
Mémoli, S. Oudot 2009
If X and Y are finite metric
spaces, the bottleneck distance between the persistence diagrams of
HRips(X) and HRips(Y ) is a
lower bound of 2dGH (X, Y ).
persistence diagrams used as
shape signatures for classification
Tame persistent modules
Definition: A persistent module is a family M = (M a , a ∈ R) of vector spaces
together with linear maps ξab : M a → M b for all a ≤ b where ξaa = idM a and
ξac = ξbc ◦ ξab for all a ≤ b ≤ c.
Examples:
1. Given a topological space X and a map f : X → R, the sublevel sets
filtration H∗ (f −1 (−∞; a]), a ∈ R, is a persistence module with linear maps
the morphisms ξab induced at the homology level by the inclusion maps
f −1 (−∞; a] ,→ f −1 (−∞; b].
2. Given a metric space (X, dX ), the Vietoris-Rips filtration HRips(X) is a persistence module with linear maps the morphisms ξab induced at the homology
level by the inclusion maps Rips(X, a) ,→ Rips(X, b).
Tame persistent modules
Definition: A persistent module is a family M = (M a , a ∈ R) of vector spaces
together with linear maps ξab : M a → M b for all a ≤ b where ξaa = idM a and
ξac = ξbc ◦ ξab for all a ≤ b ≤ c.
In the “classical” setting (finite dimentional spaces), the definition of persistence
only involves the rank of the maps ξab , a < b.
Definition: A persistence module M is tame if for any a < b, ξab has a finite rank.
Theorem [CCGGO’09]: tame persistence modules have well-defined persistence diagrams.
Tame persistent modules
Definition: A persistent module is a family M = (M a , a ∈ R) of vector spaces
together with linear maps ξab : M a → M b for all a ≤ b where ξaa = idM a and
ξac = ξbc ◦ ξab for all a ≤ b ≤ c.
A map Φ : M → N of degree t is a family of
b+t
morphisms φa : M a → N a+t for all a s.t. ηa+t
◦
φa = φb ◦ ξab .
Ma
Mb
N a+t
N b+t
An ε-interleaving between M and N is specified by two maps Φε : M → N and
Ψε : N → M of degree ε s.t. Φε ◦ Ψε and Ψε ◦ Φε are the shifts of N and M.
···
M
a
ξaa+2ε
φa
···
···
M a+2ε
ψa+ε
N a+ε
a+3ε
ηa+ε
N a+3ε
···
Tame persistent modules
Definition: A persistent module is a family M = (M a , a ∈ R) of vector spaces
together with linear maps ξab : M a → M b for all a ≤ b where ξaa = idM a and
ξac = ξbc ◦ ξab for all a ≤ b ≤ c.
A persistence module M is tame if for any a < b, ξab has a finite rank.
Stability Theorem [CCGGO’09]: If M and N are tame and ε-interleaved for some
ε ≥ 0 then
dB (Dgm(M), Dgm(N )) ≤ ε
Correspondences and interleaving
Proposition: Let X, Y be metric spaces and let C ⊂ X ×Y be an ε-correspondence
between X and Y . Then C induces a canonical ε-interleaving between HRips(X)
and HRips(Y ).
Y
C
y0
y
x
x0
X
|dX (x, x0 ) − dY (y, y 0 )| < ε
Correspondences and interleaving
Proposition: Let X, Y be metric spaces and let C ⊂ X ×Y be an ε-correspondence
between X and Y . Then C induces a canonical ε-interleaving between HRips(X)
and HRips(Y ).
Lemma: Let f : X → Y be compatible with C in the sense that (x, f (x)) ∈ C for
all x ∈ X. Then f induces a simplicial map f : Rips(X, a) → Rips(Y, b) whenever
b ≥ a + ε. Moreover, if g : X → Y is compatible with C, f , g are contiguous maps.
Y
C
y0
y
x
x0
X
|dX (x, x0 ) − dY (y, y 0 )| < ε
Correspondences and interleaving
Proposition: Let X, Y be metric spaces and let C ⊂ X ×Y be an ε-correspondence
between X and Y . Then C induces a canonical ε-interleaving between HRips(X)
and HRips(Y ).
Lemma: Let f : X → Y be compatible with C in the sense that (x, f (x)) ∈ C for
all x ∈ X. Then f induces a simplicial map f : Rips(X, a) → Rips(Y, b) whenever
b ≥ a + ε. Moreover, if g : X → Y is compatible with C, f , g are contiguous maps.
Proof of lemma:
1.
if σ = [x0 , · · · xk ] ∈ Rips(X, a) then
f (σ) = [f (x0 ), · · · f (xk )] ∈ Rips(Y, b)
2. f (σ) and g(σ) are contained in a simplex of
Rips(Y, b):
dY (f (xi ), f (xj )) ≤ dX (xi , xj ) + ε ≤ a + ε ≤ b
dY (g(xi ), g(xj )) ≤ dX (xi , xj ) + ε ≤ a + ε ≤ b
dY (f (xi ), g(xj )) ≤ dX (xi , xj ) + ε ≤ a + ε ≤ b
Y
C
y0
y
x
x0
X
|dX (x, x0 ) − dY (y, y 0 )| < ε
Correspondences and interleaving
Proposition: Let X, Y be metric spaces and let C ⊂ X ×Y be an ε-correspondence
between X and Y . Then C induces a canonical ε-interleaving between HRips(X)
and HRips(Y ).
Lemma: Let f : X → Y be compatible with C in the sense that (x, f (x)) ∈ C for
all x ∈ X. Then f induces a simplicial map f : Rips(X, a) → Rips(Y, b) whenever
b ≥ a + ε. Moreover, if g : X → Y is compatible with C, f , g are contiguous maps.
Y
C
Proof of the proposition:
- Let f : X → Y and g : Y → X be compatible with C.
y0
- C T C = {(x, x0 ) ∈ X × X|∃y ∈ Y s.t. (x, y) ∈ y
C, (x0 , y) ∈ C} is a 2ε-correspondence and h = g ◦ f
and idX are compatible functions X → X.
x
x0
X
- It follows from the lemma that h and idX are contiguous,
0
0
|d
(x,
x
)
−
d
(y,
y
)| < ε
so they induce the same maps at homology level.
X
Y
Tameness of the Rips filtration
A metric space (X, dX ) is totally bounded (or precompact) if for any ε > 0 there
exists a finite subset Fε ⊂ X such that dH (X, Fε ) < ε (i.e. ∀x ∈ X, ∃p ∈ Fε s.t.
dX (x, p) < ε).
Theorem: Let X be a totally bounded metric space. Then HRips(X) is tame.
As a consequence Dgm(HRips(X)) is well-defined!
Tameness of the Rips filtration
A metric space (X, dX ) is totally bounded (or precompact) if for any ε > 0 there
exists a finite subset Fε ⊂ X such that dH (X, Fε ) < ε (i.e. ∀x ∈ X, ∃p ∈ Fε s.t.
dX (x, p) < ε).
Theorem: Let X be a totally bounded metric space. Then HRips(X) is tame.
As a consequence Dgm(HRips(X)) is well-defined!
Proof: show that Iab : HRips(X, a) → HRips(X, b) has
finite rank whenever a < b.
Let ε = (b − a)/2 and let F ⊂ X be finite s. t.
dH (X, F ) ≤ ε/2.
Then C = {(x, f ) ∈ X × F |d(x, f ) ≤ ε} is an
ε-correspondence.
C
F
X
Tameness of the Rips filtration
A metric space (X, dX ) is totally bounded (or precompact) if for any ε > 0 there
exists a finite subset Fε ⊂ X such that dH (X, Fε ) < ε (i.e. ∀x ∈ X, ∃p ∈ Fε s.t.
dX (x, p) < ε).
Theorem: Let X be a totally bounded metric space. Then HRips(X) is tame.
As a consequence Dgm(HRips(X)) is well-defined!
Proof: show that Iab : HRips(X, a) → HRips(X, b) has
finite rank whenever a < b.
Let ε = (b − a)/2 and let F ⊂ X be finite s. t.
dH (X, F ) ≤ ε/2.
C
F
Then C = {(x, f ) ∈ X × F |d(x, f ) ≤ ε} is an
ε-correspondence.
Using the interleaving map, Iab factorizes as
HRips(X, a) → HRips(F, a + ε) → HRips(X, a + 2ε) = HRips(X, b)
finite dimensional
X
Stability of persistence diagrams
X
Y
Theorem: Let X, Y be totally bounded metric spaces. Then
db (Dgm(HRips(X)), Dgm(HRips(Y ))) ≤ 2dGH (X, Y )
Stability of persistence diagrams
X
Y
Theorem: Let X, Y be totally bounded metric spaces. Then
db (Dgm(HRips(X)), Dgm(HRips(Y ))) ≤ 2dGH (X, Y )
Proof:
For any ε > 2dGH (X, Y ) there exists an ε-correspondence C between X and Y .
C induces a canonical ε-interleaving between HRips(X) and HRips(Y ) that are tame
modules.
It follows that
db (Dgm(HRips(X)), Dgm(HRips(Y ))) ≤ ε
Stability of persistence diagrams
X
Y
Theorem: Let X, Y be totally bounded metric spaces. Then
db (Dgm(HRips(X)), Dgm(HRips(Y ))) ≤ 2dGH (X, Y )
• Robust multiscale signature for totally bounded spaces.
• A (topological) lower bound of the Gromov-Hausdorff distance.
• Same result with Cech filtrations (and some other filtrations).
• Similar results for pairs (X, f ) where f : X → R is a Lipschitz function.
• No need of the triangle inequality! → Also works for “dissimilarity measures”
Persistence-based signatures
Signatures of some elementary shapes (approximated from finite samples):
∞
1
∞
1
∞
1
0
0
0
0
geodesic
∞
1
0
0
0
0
∞
1
∞
1
∞
1
0
0
0
0
Euclidean
∞
1
0
0
0
0
Persistence-based signatures
Experimental results
:
(from C-Cohen-Steiner-Guibas-Memoli-Oudot’09)
Persistence-based signatures
:
(from C-Cohen-Steiner-Guibas-Memoli-Oudot’09)
ca
m
el
s
ca
ts
el
ep
ha
nt
s
fa
ce
s
he
ad
s
ho
rs
es
Experimental results
camels
cats
elephants
faces
heads
horses
Persistence-based signatures
Experimental results
:
(from C-Cohen-Steiner-Guibas-Memoli-Oudot’09)
0.04
camel
cat
0.03
elephant
face
head
0.02
horse
0.01
0
−0.01
−0.02
−0.03
−0.02 −0.01
0
0.01
0.02
0.03 0.04
0.05
Homework (remarks and open questions)
• Even when X is compact, Hp (Rips(X, a)), p ≥ 1, might be infinite dimensional for some value of a:
X
It is also possible to build such an example with the “open”
a Rips complex:
[x0 , x1 , · · · , xk ] ∈ Rips(X, a) ⇔ dX (xi , xj ) < a, for all i, j
Homework (remarks and open questions)
• Even when X is compact, Hp (Rips(X, a)), p ≥ 1, might be infinite dimensional for some value of a:
X
It is also possible to build such an example with the “open”
a Rips complex:
[x0 , x1 , · · · , xk ] ∈ Rips(X, a) ⇔ dX (xi , xj ) < a, for all i, j
• If X is compact, then dim H1 (Cech(X, a)) < +∞ for all a ([Smale-Smale,
C.-de Silva]).
Homework (remarks and open questions)
• Even when X is compact, Hp (Rips(X, a)), p ≥ 1, might be infinite dimensional for some value of a:
X
It is also possible to build such an example with the “open”
a Rips complex:
[x0 , x1 , · · · , xk ] ∈ Rips(X, a) ⇔ dX (xi , xj ) < a, for all i, j
• If X is compact, then dim H1 (Cech(X, a)) < +∞ for all a ([Smale-Smale,
C.-de Silva]).
• If X is geodesic, then dim H1 (Rips(X, a)) < +∞ for all a > 0 and
Dgm(H1 Rips(X)) is contained in the vertical line x = 0.
• Open problem (probably hard): for a given compact set X, how “big” can
the set S of values a such that dim Hp (Rips(X, a)) = +∞ be?
Thank You