Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway Cheeger Inequality

Nodal domain theorem for the p-Laplacian
and the related multiway Cheeger inequality
Francesco Tudisco
First virtual SIAM Imaging Science Conference
July 17, 2020
GSSI • Gran Sasso Science Institute • L’Aquila (Italy)

Based on
F. Tudisco and M. Hein, Nodal domain theorem and higher-order
Cheeger inequality for the graph p-Laplacian, EMS Journal of Spectral
Theory, 8 : 883–908 (2018)
joint work with Matthias Hein, University of Tuebingen (Germany)
1

Outline
• Motivation
• Nodal Domains of graph Laplacian
• p-Laplacian
• Cheeger inequality
2

Setting
Discrete graph
G = (V, E)
Edge weight w : E −→ +
Node weight µ : V −→ +
u
µu
v
µv
wu,v
unoriented, connected, finite, simple
In this talk: µ(u) = 1 and w(uv) = 1 for all u ∈ V, uv ∈ E
3

Motivation
Graph clustering
G = (V, E), V = {u1,...,un}, E ⊆ V × V 4

Balanced cut problem
S
S
cut(S)
hG(2) = min
S⊆V
cut(S)
min{|S|,|S|}
= min
{S1,S2}
disjoint
max
i=1,2
cut(Si)
|Si|
cut(S) = {uv ∈ E : u ∈ S, v ∈ S}, S = V S
5

k-th Cheeger constant
S
S2
cut(S1)
S3
cut(S2)
cut(S3)
hG(k) = min
{S1,...,Sk}
dis joint
max
i=1,...,k
cut(Si)
|Si|
Relax this problem using the graph Laplacian
f −→ L f (u) =
v∼u
f (u) − f (v) L = D − A = B B
6

Laplacian relaxation
hG(k) =
1
2
min
fi ∈ {0,1}n
f1⊥···⊥fk
max
i=1,...,k
fi
L fi
fi
2
2
≥
1
2
min
X ⊆ V
dim(X) ≥ k
max
f ∈X
f L f
f 2
2
= λk(L)
7

Laplacian relaxation
hG(k) =
1
2
min
fi ∈ {0,1}n
f1⊥···⊥fk
max
i=1,...,k
fi
L fi
fi
2
2
≥
1
2
min
X ⊆ V
dim(X) ≥ k
max
f ∈X
f L f
f 2
2
= λk(L)
Approximation: use λk(L) and the associated eigenvectors
The quality of this approximation can be studies using the nodal
domains of such eigenvectors
7

Strong Nodal domains
Strong Nodal Domains of f ∈ V
= maximal connected compo-
nents in G({u : f (u) > 0}) and G({u : f (u) < 0})
ν(f ) = number of strong nodal domains
8

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
f (1)
f (2)
f (3)
f (4)
f (5)
f (6)
f (7)
f (8)
f (9)
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8

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
f (1)
f (2)
f (3)
f (4)
f (5)
f (6)
f (7)
f (8)
f (9)

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+
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−
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
ν(f ) = 3
The domains are
disjoint
8

Weak nodal domains
Weak Nodal Domains of f ∈ V
nents in G({u : f (u) ≥ 0}) and G({u : f (u) ≤ 0})
ν0(f ) = number of weak nodal domains
9

Weak nodal domains















f (1)
f (2)
f (3)
f (4)
f (5)
f (6)
f (7)
f (8)
f (9)



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

=
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
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


+
+
+
0
−
−
0
−
−
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9

Weak nodal domains















f (1)
f (2)
f (3)
f (4)
f (5)
f (6)
f (7)
f (8)
f (9)



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
=
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



+
+
+
0
−
−
0
−
−


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


ν0(f ) = 2
The domains can
overlap
9

Nodal domains theorem for L
0 = λ1(L) < λ2(L) ≤ λ3(L) ≤ ··· ≤ λn(L)
10

Nodal domains theorem for L
0 = λ1(L) < λ2(L) ≤ λ3(L) ≤ ··· ≤ λn(L)
E. Davies, G. Gladwell, J. Leydold, P. Stadler, 2000
Theorem
Let fk be an eigenvector associated to λk(L). Then
• ν(fk) ≤ k + mult(λk) − 1
• ν0(fk) ≤ k
where mult(λk) = multiplicity of λk(L)
10

The p-Laplacian
For p > 1 define
f −→ Lp f (u) =
v∼u
|f (u) − f (v)|p−2
(f (u) − f (v)) Lp(f ) = B Φp(B f )
where Φp g(u) = |g(u)|p−2
g(u)
11

The p-Laplacian
For p > 1 define
f −→ Lp f (u) =
v∼u
|f (u) − f (v)|p−2
(f (u) − f (v)) Lp(f ) = B Φp(B f )
where Φp g(u) = |g(u)|p−2
g(u)
Eigenvalues and eigenfunctions of Lp:
• critical values/points of R(f ) =
f Lp f
f
p
p
or, equivalently,
• solution of the equation Lp f (u) = λΦp f (u)
11

Variational characterization
Linear case:
the eigenvalues and eigenfunctions of L can be characterized as
Yk = {subspaces of dimension ≥ k}
λk(L) = min
Y ∈Yk
max
f ∈ Y
f L f
f 2
2
12

Variational characterization
Linear case:
the eigenvalues and eigenfunctions of L can be characterized as
Yk = {subspaces of dimension ≥ k}
λk(L) = min
Y ∈Yk
max
f ∈ Y
f L f
f 2
2
For p = 2 we can do something similar
12

Krasnoselskii genus
Let Sp = {f ∈ V
: f p = 1}
13

Krasnoselskii genus
Let Sp = {f ∈ V
: f p = 1}
Let Y = −Y be a closed symmetric set in Sp. Define
γ(Y ) = inf{m | ∃h : Y → m
{0}, continuous, h(−u) = −h(u)}
13

Krasnoselskii genus
Let Sp = {f ∈ V
: f p = 1}
Let Y = −Y be a closed symmetric set in Sp. Define
γ(Y ) = inf{m | ∃h : Y → m
{0}, continuous, h(−u) = −h(u)}
γ is a form of dimension of Y :
• Y = bounded symm neighborhood of 0 in k
=⇒ γ(Y ) = k.
• Viceversa: if γ(Y ) = k =⇒ Y contains at least k orthogonal vectors.
13

Variational spectrum of Lp
For any integer k, consider the family
Yk = {Y ⊆ Sp, closed, symmetric, s.t. γ(Y ) ≥ k}
14

Variational spectrum of Lp
For any integer k, consider the family
Yk = {Y ⊆ Sp, closed, symmetric, s.t. γ(Y ) ≥ k}
Theorem [Lusternik-Schnirelman]
The numbers
λk(Lp) = min
Y ∈Yk
max
f ∈ Y
f Lp f
f
p
p
are n (variational) eigenvalues of Lp.
14

Two remarks
• λ1(Lp) = 0 and corresponding eigenfunctions are constant
• multiplicity of λ1(Lp) = # connected components, i.e.
λn(Lp) ≥ ··· ≥ λ2(Lp) > λ1(Lp) = 0
λk(Lp) has multiplicity m means:
λk(Lp) = λk+1(Lp) = ··· = λk+m−1(Lp)
15

Nodal domain theorem p > 1
Theorem [T. & Hein]
Let fk be any eigenfunction associated to λk(Lp). Then
• ν(fk) ≤ k + mult(λk) − 1
• ν0(uk) ≤ k
16

Tightness for path graphs
When G is a path
we have
Theorem
• mult(λk(Lp)) = 1 for all k = 1,..., n
• for any associated eigenvector ν0(fk) = ν(fk) = k
17

Nodal domain theorem p = 1
The proof for ν0(f ) that we have does not work when p = 1
18

The definition changes into:
L1 f (u) = v∈V wu,vzu,v : zu,v = −zv,u, zu,v ∈ Sign(f (u) − f (v))
where Sign(x) = ±1, if x = 0 and Sign(0) = [−1,1]
18

The definition changes into:
L1 f (u) = v∈V wu,vzu,v : zu,v = −zv,u, zu,v ∈ Sign(f (u) − f (v))
where Sign(x) = ±1, if x = 0 and Sign(0) = [−1,1]
There are examples such that ν0(fk) = k + mult(λk) − 1 thus,
Theorem
For the 1-Laplacian it holds:
ν(fk),ν0(fk) ≤ k + mult(λk) − 1
18

k-th Cheeger constant: relaxation with Lp
hG(k) = min
{S1,...,Sk}
disjoint
max
i=1,...,k
cut(Si)
|Si|
19

hG(k) = min
{S1,...,Sk}
disjoint
max
i=1,...,k
cut(Si)
|Si|
=
1
2p−1
min
fi ∈ {0,1}V
f1⊥···⊥fk
max
i=1,...,k
fi
Lp fi
fi
p
p
19

hG(k) = min
{S1,...,Sk}
disjoint
max
i=1,...,k
cut(Si)
|Si|
=
1
2p−1
min
fi ∈ {0,1}V
f1⊥···⊥fk
max
i=1,...,k
fi
Lp fi
fi
p
p
≥
1
2p−1
min
Y ∈Yk
max
f ∈ Y
f Lp f
f
p
p
= λk(Lp)
19

hG(k) = min
{S1,...,Sk}
disjoint
max
i=1,...,k
cut(Si)
|Si|
=
1
2p−1
min
fi ∈ {0,1}V
f1⊥···⊥fk
max
i=1,...,k
fi
Lp fi
fi
p
p
≥
1
2p−1
min
Y ∈Yk
max
f ∈ Y
f Lp f
f
p
p
= λk(Lp)
Cheeger inequality: how good is the approximation?
19

Cheeger inequality
Theorem [T. & Hein]
Let fk be any eigenfunction of λk(Lp) and let m = ν(fk),
then:
2
τ(G)
p−1
hG(m)
p
p
≤ λk(Lp) ≤ 2p−1
hG(k)
where τ(G) = maxu degree(u)
20

Consequences for k = 2
Let f be any eigenvector of λ2(Lp). Then
• The graphs G({u : f (u) > 0}) and G({u : f (u) < 0}) are connected
• λ2(Lp) −→ h2(G) as p −→ 1 [Amghibech 03, Buehler&Hein 09]
21

Back to the initial picture
λ2(L2) ≈ h2(G)
p = 2
22

Back to the initial picture
λ2(L2) ≈ h2(G)
p = 2
p −→ 1
λ2(Lp) = h2(G)
p = 1
22

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