A new Perron-Frobenius theorem for nonnegative tensors

A new Perron–Frobenius theorem for
nonnegative tensors
Francesco Tudisco
joint work with Antoine Gautier and Matthias Hein (Saarland University)
Department of Mathematics and Statistics, University of Strathclyde, UK
SIAM ALA Conference • Hong Kong Baptist University
May 5, 2018
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Outline
Huge literature on eigenvalues of nonnegative tensors
[Bai, Caccetta, Chang, Friedland, Gaubert, Han, Hu, Huang,
Kang, Kolda, Lim, Ling, Liu, Mayo, Ng, Pearson, Ottaviani, Qi,
Sun, Xie, Yang, You, Yuan, Wang, Zhang, Zhou, ...]
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Outline
Huge literature on eigenvalues of nonnegative tensors
[Bai, Caccetta, Chang, Friedland, Gaubert, Han, Hu, Huang,
Kang, Kolda, Lim, Ling, Liu, Mayo, Ng, Pearson, Ottaviani, Qi,
Sun, Xie, Yang, You, Yuan, Wang, Zhang, Zhou, ...]
1. Shape partitions:
A unifying eigenvalue problem formulation for tensors
2. Homogeneity matrix:
Perron-Frobenius thm for tensors via multi-homogeneous map
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Notation
T ∼ N1 × · · · × Nm
T = (ti1,...,im ) ∈ RN1×···×Nm
+ nonnegative tensor
x = (x1, · · · , xm) ∈ RN1 × · · · × RNm
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Notation
Two useful maps
T ∼ N1 × · · · × Nm
T : RN1 × · · · × RNm −→ R
T(x) =
i1,...,im
ti1,...,im x1,i1 · · · xm,im
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Notation
Two useful maps
T ∼ N1 × · · · × Nm
T : RN1 × · · · × RNm −→ R
T(x) =
i1,...,im
ti1,...,im x1,i1 · · · xm,im
T[k] : RN1 × · · · × RNm −→ RNk
T[k](x)ik
=
i1,...,ik−1,ik+1,...,im
ti1,...,ik,...,im x1,i1 · · · xk−1,ik−1
xk+1,ik+1
· · · xm,im
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Notation
Two useful maps
T ∼ N1 × · · · × Nm
T : RN1 × · · · × RNm −→ R
T(x) =
i1,...,im
ti1,...,im x1,i1 · · · xm,im
T[k] : RN1 × · · · × RNm −→ RNk
T[k](x)ik
=
i1,...,ik−1,ik+1,...,im
ti1,...,ik,...,im x1,i1 · · · xk−1,ik−1
xk+1,ik+1
· · · xm,im
Example: Matrix case
T ∼ N × N, i.e. T = matrix A
T(x) = xTAx T[1](x) = Ax T[2](x) = ATx
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Example m = 3 and N1 = N2 = N3
Eigenvalue problems
T ∼ N1 × N1 × N1, square tensor
f1 =
T(x, x, x)
x 3
p
f2 =
T(x, y, y)
x p y 2
q
f3 =
T(x, y, z)
x p y q z r
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Eigenvalue problems
f1 =
T(x, x, x)
x 3
p
ℓp-eigenvalues
If T is symmetric (tijk = tikj = tkij = tkji)
Crit. points f1 ⇐⇒ T[1](x, x, x) = λ xp−2
x
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Eigenvalue problems
f2 =
T(x, y, y)
x p y 2
q
ℓp,q-singular values
If T is partially symmetric (tijk = tikj)
Crit. points f2 ⇐⇒
T[1](x, y, y) = λ xp−2x
T[2](x, y, y) = λ yq−2y
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Eigenvalue problems
f3 =
T(x, y, z)
x p y q z r
ℓp,q,r-singular values
For any T
Crit. points f3 ⇐⇒



T[1](x, y, z) = λ xp−2x
T[2](x, y, z) = λ yq−2y
T[3](x, y, z) = λ zr−2z
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Eigenvalue problems
We introduce a unifying formulation
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Shape partition
Shape partition of T ∼ N1 × · · · × Nm:
σ = {σ1, . . . , σd} is a partition of {1, . . . , m} s.t.
• j, k ∈ σi =⇒ Ni = Nk
• j ∈ σi, k ∈ σi+1 =⇒ j < k
• |σi| ≤ |σi+1|
Examples:
T ∼ N × N × N × N
σ1 = {1, 2, 3, 4}
σ2 = {1}, {2, 3, 4}
σ3 = {1, 2}, {3, 4}
σ4 = {1}, {2}, {3, 4}
σ5 = {1}, {2}, {3}, {4}
T ∼ N × N × M × M
σ3 = {1, 2}, {3, 4}
σ4 = {1}, {2}, {3, 4}
σ5 = {1}, {2}, {3}, {4}
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Indexing associated with σ
T ∼ N1 × · · · × Nm
σ = {σ1, . . . , σd}
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Indexing associated with σ
T ∼ N1 × · · · × Nm
σ = {σ1, . . . , σd}
• si = min{k : k ∈ σi}
• ni = Nsi for all i = 1, . . . , d
σ1
N1 × . . . × Ns2−1
=
=
n1 × . . . × n1
|σ1| times
×
=
×
σ2
Ns2 × . . . × Ns3−1
=
=
n2 × . . . × n2
|σ2| times
× . . . ×
=
=
× . . . ×
σd
Nsd × . . . × Nm
=
=
nd × . . . × nd
|σd| times
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σ−eigenvalues and eigenvectors
T ∼ N1 × · · · × Nm, σ = {σ1, . . . , σd}, p = (p1, . . . , pd) > 1
|σ1| times
x1, . . . , x1,
|σ2| times
x2, . . . , x2, . . . ,
|σd| times
xd, . . . , xd
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T ∼ N1 × · · · × Nm, σ = {σ1, . . . , σd}, p = (p1, . . . , pd) > 1
|σ1| times
x1, . . . , x1,
|σ2| times
x2, . . . , x2, . . . ,
|σd| times
xd, . . . , xd
(⋆)



T[s1](x1,..., x1, . . . , xd,..., xd) = λ xp1−2
1 x1 x1 p1 = 1
...
T[sd](x1,..., x1, . . . , xd,..., xd) = λ xpd−2
d xd xd pd
= 1
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T ∼ N1 × · · · × Nm, σ = {σ1, . . . , σd}, p = (p1, . . . , pd) > 1
|σ1| times
x1, . . . , x1,
|σ2| times
x2, . . . , x2, . . . ,
|σd| times
xd, . . . , xd
(⋆)



T[s1](x1,..., x1, . . . , xd,..., xd) = λ xp1−2
1 x1 x1 p1 = 1
...
T[sd](x1,..., x1, . . . , xd,..., xd) = λ xpd−2
d xd xd pd
= 1
(T, σ, p) −→ r(T) = max{|λ| : λ fulfill (⋆)}
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Multi-homogeneous map associated to (T, σ, p)
We characterize the existence, uniqueness and maximality of
positive σ−eigenpairs of T.
To this end we consider a particular multi-homogeneous map
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F = (f1, . . . , fm) : RN1 × · · · × RNm −→ RN1 × · · · × RNm s.t.
fi(x1, . . . , λ xj, . . . , xm) = λAij
fi(x1, . . . , xj, . . . , xm)
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F = (f1, . . . , fm) : RN1 × · · · × RNm −→ RN1 × · · · × RNm s.t.
fi(x1, . . . , λ xj, . . . , xm) = λAij
fi(x1, . . . , xj, . . . , xm)
A = A(F) = homogeneity matrix of F
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Given T, σ and p we define FT,σ,p = (f1, . . . , fd) by
fi(x) = T[si]
|σ1| times
x1, . . . , x1, . . . ,
|σd| times
xd, . . . , xd
1
pi−1

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Given T, σ and p we define FT,σ,p = (f1, . . . , fd) by
fi(x) = T[si]
|σ1| times
x1, . . . , x1, . . . ,
|σd| times
xd, . . . , xd
1
pi−1
A =












|σ1|−1
p1−1
|σ2|
p1−1 · · · |σd|
p1−1
|σ1|
p2−1
|σ2|−1
p2−1 · · · |σd|
p2−1
... ... ... ...
|σ1|
pd−1
|σ2|
pd−1 · · · |σd|−1
pd−1












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Perron-Frobenius theorem for “contractive” (T, σ, p)
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Theorem
If ρ(A) < 1 and T is σ-strictly nonnegative
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σ-strictly nonnegative
For every (i, ki) ∈ Iσ there exist j1, . . . , jm such that:
tj1,...,jm > 0 with jsi = ki
Theorem
If ρ(A) < 1 and T is σ-strictly nonnegative
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Theorem
If ρ(A) < 1 and T is σ-strictly nonnegative, then:
• There exists a entry-wise positive σ-eigenvector u of T
associated with r(T)
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Theorem
If ρ(A) < 1 and T is σ-strictly nonnegative, then:
• There exists a entry-wise positive σ-eigenvector u of T
associated with r(T)
• u is the unique positive σ-eigenvector of T
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Example: ℓp,q,r
-singular value problem
T ∼ N1 × N2 × N3, σ = {{1}, {2}, {3}}, p = (p, q, r)
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Example: ℓp,q,r
T ∼ N1 × N2 × N3, σ = {{1}, {2}, {3}}, p = (p, q, r)



T[1](x, y, z) = λ xp−2x
T[2](x, y, z) = λ yq−2y
T[3](x, y, z) = λ zr−2z
(⋆⋆)
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Example: ℓp,q,r
T ∼ N1 × N2 × N3, σ = {{1}, {2}, {3}}, p = (p, q, r)



T[1](x, y, z) = λ xp−2x
T[2](x, y, z) = λ yq−2y
T[3](x, y, z) = λ zr−2z
(⋆⋆)
There exists a unique positive u = (x, y, z) that solves (⋆⋆) with
λ = r(T) if:
• ρ(A) < 1 and strictly nonnegative
[Gautier, T., Hein, 2018]
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Example: ℓp,q,r
T ∼ N1 × N2 × N3, σ = {{1}, {2}, {3}}, p = (p, q, r)



T[1](x, y, z) = λ xp−2x
T[2](x, y, z) = λ yq−2y
T[3](x, y, z) = λ zr−2z
(⋆⋆)
There exists a unique positive u = (x, y, z) that solves (⋆⋆) with
λ = r(T) if:
• ρ(A) < 1 and strictly nonnegative
[Gautier, T., Hein, 2018]
• p, q, r ≥ 3 and irreducibility conditions
[Friedland, Gaubert, Han, 2013]
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The “non-exapansive” case ρ(A) = 1
T ∼ N × N, σ = {1, 2}, p = 2 ⇐⇒ Tx = λx

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The “non-exapansive” case ρ(A) = 1
T ∼ N × N, σ = {1, 2}, p = 2 ⇐⇒ Tx = λx
A =
|σ1| − 1
p1 − 1
= 1
We know that we need irreducibility conditions
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σ−graph of T
Gσ(T) = (V, E), σ = {σ1, . . . , σd}
• Nodes are pairs:
V = {1} × {1, . . . , n1} ∪ · · · ∪ {d} × {1, . . . , nd}

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σ−graph of T
Gσ(T) = (V, E), σ = {σ1, . . . , σd}
V = {1} × {1, . . . , n1} ∪ · · · ∪ {d} × {1, . . . , nd}
• Edge (ik) → (jh) exists if
tℓ1,...,ℓm > 0 with ℓsi = k and ℓα = h for some α ∈ σi
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σ−graph of T
Gσ(T) = (V, E), σ = {σ1, . . . , σd}
V = {1} × {1, . . . , n1} ∪ · · · ∪ {d} × {1, . . . , nd}
Example:
T ∼ 2 × 3 × 3
σ = {{1}, {2, 3}}, d = 2
n1 = 2, s1 = 1, n2 = 3, s2 = 2
1211 {1}
2221 23 {2,3}
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σ−graph of T
Gσ(T) = (V, E), σ = {σ1, . . . , σd}
V = {1} × {1, . . . , n1} ∪ · · · ∪ {d} × {1, . . . , nd}
Example:
T ∼ 2 × 3 × 3
σ = {{1}, {2, 3}}, d = 2
n1 = 2, s1 = 1, n2 = 3, s2 = 2
1211 {1}
2221 23 {2,3}
t2,2,1 = 1
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σ−graph of T
Gσ(T) = (V, E), σ = {σ1, . . . , σd}
V = {1} × {1, . . . , n1} ∪ · · · ∪ {d} × {1, . . . , nd}
Example:
T ∼ 2 × 3 × 3
σ = {{1}, {2, 3}}, d = 2
n1 = 2, s1 = 1, n2 = 3, s2 = 2
1211 {1}
2221 23 {2,3}
t2,2,1 = 1 t1,3,1 = 1
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Perron-Frobenius thm for “non-expansive” (T, σ, p)
Theorem
If ρ(A) ≤ 1 and T is σ-weakly irreducible
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σ-weakly irreducible
The graph Gσ(T) is strongly connected
Theorem
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σ-weakly irreducible =⇒ σ-strictly nonnegative
Theorem
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σ-weakly irreducible =⇒ σ-strictly nonnegative
Theorem
If ρ(A) ≤ 1 and T is σ-weakly irreducible, then additionally:
• It holds λ < r(T) for every σ-eigenvalue λ
corresponding to a nonnegative eigenvector
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σ−strongly irreducibile T
G(M) is strongly connected if and only if for any x0 ≥ 0 there
exists n > 0 such that xn = (I + M)nx0 > 0
xn = xn−1 + Mxn−1 = xn−1 + Mxn−2 + M2xn−2 = · · ·
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F = FT,σ,p = multi-homogeneous map associated with (T, σ, p)
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F = FT,σ,p = multi-homogeneous map associated with (T, σ, p)
σ-strongly irreducible
For any x0 ≥ 0 there exists n > 0 s.t. xn = (id + F)n(x0) > 0
xn = xn−1 + F(xn−1) = xn−1 + F(xn−2 + F(xn−2)) = · · ·
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Perron-Frobenius thm for σ-strongly irreducible T
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Theorem
σ-strongly irreducible =⇒ σ-weakly irreducible
(=⇒ σ-strictly nonnegative)
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Theorem
Theorem
If ρ(A) ≤ 1 and T is σ-strongly irreducible, then additionally:
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Theorem
Theorem
If ρ(A) ≤ 1 and T is σ-strongly irreducible, then additionally:
• T has no nonnegative eigenvectors other than u
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A final remark:
Previous conditions for particular choices of σ
• σ-strictly nonnegativity for σ = {1, . . . , m}
[Hu, Huang, Qi, 2014]
• σ-weak irreducibility and σ-strong irreducibility for
σ = {1, . . . , m}
σ = {1, . . . , k}, {k + 1, . . . , m}
σ = {1}, . . . , {m}
[Chang, Pearson, Zhang, 2008] & [Chang, Qi, Zhou, 2010] &
[Friedland, Gaubert, Han, 2013] & [Ling, Qi, 2013]
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Conclusion
• A large class of eigenvalue problems for tensors can be written
in terms of (T, σ ,p)
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Conclusion
• If T ≥ 0 and it does not have fully zero unfoldings, then look
at the matrix A = A(FT,σ,p):
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Conclusion
• If ρ(A) < 1, then PF holds
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Conclusion
• If ρ(A) = 1 and Gσ(T) is connected,
then PF holds + r(T) > λ
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Conclusion
• If ρ(A) = 1 and Gσ(T) is connected,
then PF holds + r(T) > λ
• If id + FT,σ,p is “eventually positive”,
then unique nonnegative eigenvector
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References
A. Gautier, F. T. and M. Hein,
A unifying Perron-Frobenius theorem for nonnegative tensors
via multi-homogeneous maps, SIAM J Matrix Anal Appl 2019
A. Gautier, F. T. and M. Hein,
The Perron-Frobenius theorem for multi-homogeneous
mapping, SIAM J Matrix Anal Appl 2019
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Thank you for your kind attention!
MS20
Nonlinear Perron-Frobenius theory and applications
jointly organized with Antoine Gautier
Today
4:15PM – 6:15PM
WLB103
Speakers:
• Antoine Gautier
• Francesca Arrigo
• Shmuel Friedland
• Jiang Zhou
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A new Perron-Frobenius theorem for nonnegative tensors

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