Based on the concept of dimensional partition we consider a general tensor spectral problem that includes all known tensor spectral problems as special cases. We formulate irreducibility and symmetry properties in terms of the dimensional partition and use the theory of multi-homogeneous order-preserving maps to derive a general and unifying Perron-Frobenius theorem for nonnegative tensors that either includes previous results of this kind or improves them by weakening the assumptions there considered.
Talk presented at SIAM Applied Linear Algebra conference Hong Kong 2018
A new Perron-Frobenius theorem for nonnegative tensors
1. 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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7. 3/20
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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Example m = 3 and N1 = N2 = N3
Eigenvalue problems
T ∼ N1 × N1 × N1, square tensor
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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Example m = 3 and N1 = N2 = N3
Eigenvalue problems
T ∼ N1 × N1 × N1, square tensor
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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Example m = 3 and N1 = N2 = N3
Eigenvalue problems
T ∼ N1 × N1 × N1, square tensor
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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Example m = 3 and N1 = N2 = N3
Eigenvalue problems
T ∼ N1 × N1 × N1, square tensor
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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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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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
F = (f1, . . . , fm) : RN1 × · · · × RNm −→ RN1 × · · · × RNm s.t.
fi(x1, . . . , λ xj, . . . , xm) = λAij
fi(x1, . . . , xj, . . . , xm)
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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
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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Multi-homogeneous map associated to (T, σ, p)
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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Perron-Frobenius theorem for “contractive” (T, σ, p)
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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27. 10/20
Perron-Frobenius theorem for “contractive” (T, σ, p)
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, then:
• There exists a entry-wise positive σ-eigenvector u of T
associated with r(T)
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28. 10/20
Perron-Frobenius theorem for “contractive” (T, σ, p)
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, 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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The “non-exapansive” case ρ(A) = 1
Example: Matrix case
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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Perron-Frobenius thm for “non-expansive” (T, σ, p)
σ-weakly irreducible
The graph Gσ(T) is strongly connected
Theorem
If ρ(A) ≤ 1 and T is σ-weakly irreducible
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Perron-Frobenius thm for “non-expansive” (T, σ, p)
σ-weakly irreducible
The graph Gσ(T) is strongly connected
σ-weakly irreducible =⇒ σ-strictly nonnegative
Theorem
If ρ(A) ≤ 1 and T is σ-weakly irreducible
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Perron-Frobenius thm for “non-expansive” (T, σ, p)
σ-weakly irreducible
The graph Gσ(T) is strongly connected
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
Example: Matrix case
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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σ−strongly irreducibile T
Example: Matrix case
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 = · · ·
F = FT,σ,p = multi-homogeneous map associated with (T, σ, p)
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σ−strongly irreducibile T
Example: Matrix case
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 = · · ·
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
Theorem
σ-strongly irreducible =⇒ σ-weakly irreducible
(=⇒ σ-strictly nonnegative)
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Perron-Frobenius thm for σ-strongly irreducible T
Theorem
σ-strongly irreducible =⇒ σ-weakly irreducible
(=⇒ σ-strictly nonnegative)
Theorem
If ρ(A) ≤ 1 and T is σ-strongly irreducible, then additionally:
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Perron-Frobenius thm for σ-strongly irreducible T
Theorem
σ-strongly irreducible =⇒ σ-weakly irreducible
(=⇒ σ-strictly nonnegative)
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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Conclusion
• A large class of eigenvalue problems for tensors can be written
in terms of (T, σ ,p)
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Conclusion
• A large class of eigenvalue problems for tensors can be written
in terms of (T, σ ,p)
• 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
• A large class of eigenvalue problems for tensors can be written
in terms of (T, σ ,p)
• If T ≥ 0 and it does not have fully zero unfoldings, then look
at the matrix A = A(FT,σ,p):
• If ρ(A) < 1, then PF holds
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Conclusion
• A large class of eigenvalue problems for tensors can be written
in terms of (T, σ ,p)
• If T ≥ 0 and it does not have fully zero unfoldings, then look
at the matrix A = A(FT,σ,p):
• If ρ(A) < 1, then PF holds
• If ρ(A) = 1 and Gσ(T) is connected,
then PF holds + r(T) > λ
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Conclusion
• A large class of eigenvalue problems for tensors can be written
in terms of (T, σ ,p)
• If T ≥ 0 and it does not have fully zero unfoldings, then look
at the matrix A = A(FT,σ,p):
• If ρ(A) < 1, then PF holds
• 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
20 / 20
![1/20
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 / 20](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)