The Miracles of Tropical Spectral Theory

The Miracles of Tropical Spectral Theory Emmanuel Tsukerman University of California, Berkeley e.tsukerman@berkeley.edu July 31, 2015 1 / 110

Motivation One of the most important results in max-algebra [is] that for every matrix the maximum cycle mean is the greatest eigenvalue [But10] 2 / 110

Motivation One of the most important results in max-algebra [is] that for every matrix the maximum cycle mean is the greatest eigenvalue [But10] - Tensor spectral theory is flourishing, so let us study its tropicalization 3 / 110

Outline 1 Preliminaries Basic Tropical Geometry Tropical Spectral Theory of Matrices Classical Tensor Eigenpairs 2 Tropical Spectral Theory of Tensors 5 / 110

Outline 1 Preliminaries Basic Tropical Geometry Tropical Spectral Theory of Matrices Classical Tensor Eigenpairs 2 Tropical Spectral Theory of Tensors 6 / 110

Basic Tropical Geometry 7 / 110

Tropical Arithmetic Replace ordinary addition and multiplication with: x y = x y = minimum of x and y x + y 8 / 110

Tropical Arithmetic Replace ordinary addition and multiplication with: x y = x y = minimum of x and y x + y Example: 3 (4 5) = 3 4 = 7 9 / 110

Tropical Arithmetic Replace ordinary addition and multiplication with: x y = x y = minimum of x and y x + y Example: 3 (4 5) = 3 4 = 7 3 (4 5) = 3 4 3 5 = 7 8 = 7 10 / 110

Tropical Arithmetic Replace ordinary addition and multiplication with: x y = x y = minimum of x and y x + y Example: 3 (4 5) = 3 4 = 7 3 (4 5) = 3 4 3 5 = 7 8 = 7 Associative, distributive, commutative 11 / 110

Tropical Arithmetic There is also an additive identity 12 / 110

Tropical Arithmetic There is also an additive identity x = x for any x 13 / 110

Tropical Arithmetic Matrix times vector multiplication: 14 / 110

Tropical Arithmetic Matrix times vector multiplication: ( ) ( ) ( a11 a 12 x a11 x a = 12 y a 21 a 22 y... ) 15 / 110

Tropical Arithmetic Matrix times vector multiplication: ( ) ( ) ( a11 a 12 x a11 x a = 12 y a 21 a 22 y... ( ) min(a11 + x, a = 12 + y)... ) 16 / 110

Tropical Arithmetic Matrix times vector multiplication: ( ) ( ) ( a11 a 12 x a11 x a = 12 y a 21 a 22 y... Scalar times matrix multiplitication: ( ) min(a11 + x, a = 12 + y)... ) 17 / 110

Tropical Arithmetic Matrix times vector multiplication: ( ) ( ) ( a11 a 12 x a11 x a = 12 y a 21 a 22 y... ( ) min(a11 + x, a = 12 + y)... Scalar times matrix multiplitication: ( ) ( ) a11 a λ 12 λ a11 λ a = 12 a 21 a 22 λ a 21 λ a 22 ) 18 / 110

Outline 1 Preliminaries Basic Tropical Geometry Tropical Spectral Theory of Matrices Classical Tensor Eigenpairs 2 Tropical Spectral Theory of Tensors 20 / 110

Tropical Spectral Theory of Matrices 21 / 110

Tropical Eigenpair An eigenvalue of an n n matrix A is a number λ such that A v = λ v 22 / 110

Tropical Eigenpair An eigenvalue of an n n matrix A is a number λ such that A v = λ v A vector v as above is called an eigenvector. 23 / 110

Tropical Eigenpair An eigenvalue of an n n matrix A is a number λ such that A v = λ v A vector v as above is called an eigenvector. Example: n = 2 ( ) min(a11 + x, a 12 + y) = min(a 21 + x, a 22 + y) ( λ + x λ + y ) 24 / 110

Directed Graph of a Square Matrix Associate to an n n matrix A = (a ij ) a weighted directed graph G(A) as follows. The nodes are 1, 2,..., n 25 / 110

Directed Graph of a Square Matrix Associate to an n n matrix A = (a ij ) a weighted directed graph G(A) as follows. The nodes are 1, 2,..., n There is an edge from i to j iff a ij <. 26 / 110

Directed Graph of a Square Matrix Associate to an n n matrix A = (a ij ) a weighted directed graph G(A) as follows. The nodes are 1, 2,..., n There is an edge from i to j iff a ij <. Edge (i, j) gets weight a ij A directed path i 0, i 1,..., i k gets a weight: The normalized weight a i0 i 1 + a i1 i 2 +... + a ik 1 i k. a i0 i 1 + a i1 i 2 +... + a ik 1 i k k is called the normalized length of the path. 29 / 110

Tropical Eigenvalue and the Directed Graph Theorem [CG62, Cuninghame-Green] If the graph G(A) is strongly connected, then A has precisely one eigenvalue. This eigenvalue is equal to the minimal normalized length of any directed cycle of G(A). It appears then that in order to find λ(a) we must inspect all cycles of G(A) 32 / 110

Tropical Eigenvalue via LP λ(a) = [ maximize λ subject to a ij + x j λ + x i, i, j [n] ] 34 / 110

Tropical Eigenvalue via LP λ(a) = [ maximize λ subject to a ij + x j λ + x i, i, j [n] This is surprising because the constraints are obvious necessary conditions and no equalities are enforced Example: n = 2 min(a 11 + x 1, a 12 + x 2 ) = λ + x 1 min(a 21 + x 1, a 22 + x 2 ) = λ + x 2 ] 36 / 110

Classical and Tropical Eigenvalues 37 / 110

Classical and Tropical Eigenvalues Let X be a positive matrix over a field with valuation. 38 / 110

Classical and Tropical Eigenvalues Let X be a positive matrix over a field with valuation. Let X (k) be its kth Hadamard power. 39 / 110

Classical and Tropical Eigenvalues Let X be a positive matrix over a field with valuation. Let X (k) be its kth Hadamard power.let ρ denote the Perron-Frobenius eigenvalue and λ the tropical eigenvalue. 40 / 110

Outline 1 Preliminaries Basic Tropical Geometry Tropical Spectral Theory of Matrices Classical Tensor Eigenpairs 2 Tropical Spectral Theory of Tensors 42 / 110

Classical Tensor Eigenpairs 43 / 110

Tensor times a vector Take A = (a i1 i 2 i m ) to be an } n. {{.. n } tensor. Let x be an n-vector. m times 44 / 110

Tensor times a vector Take A = (a i1 i 2 i m ) to be an n... n }{{} We define the product Ax m 1 := m times tensor. Let x be an n-vector. n i 2,...,i m=1 a 1i 2 i m x i2 x im n i 2,...,i m=1 a 2i 2 i m x i2 x im. n i 2,...,i m=1 a ni 2 i m x i2 x im. 45 / 110

Tensor eigenpairs There are several variants of eigenpairs: 46 / 110

Tensor eigenpairs There are several variants of eigenpairs: x m 1 1 Ax m 1 = λx [m 1] x m 1 = λ 2. xn m 1 (H-eigenpair) 47 / 110

Tensor eigenpairs There are several variants of eigenpairs: x m 1 1 Ax m 1 = λx [m 1] x m 1 = λ 2. xn m 1 Ax m 1 = λx (H-eigenpair) (E-eigenpair) 48 / 110

Tensor eigenpairs There are several variants of eigenpairs: x m 1 1 Ax m 1 = λx [m 1] x m 1 = λ 2. xn m 1 and others which I will not discuss. Ax m 1 = λx (H-eigenpair) (E-eigenpair) 49 / 110

Outline 1 Preliminaries Basic Tropical Geometry Tropical Spectral Theory of Matrices Classical Tensor Eigenpairs 2 Tropical Spectral Theory of Tensors 50 / 110

Tropical Spectral Theory of Tensors 51 / 110

Tropical Eigenpairs The tropicalization of Ax m 1 := n i 2,...,i m=1 a 1i 2 i m x i2 x im n i 2,...,i m=1 a 2i 2 i m x i2 x im. n i 2,...,i m=1 a ni 2 i m x i2 x im should be n i 2,...,i m=1 a 1i 2 i m x i2 x im n i 2,...,i m=1 a 2i 2 i m x i2 x im. n i 2,...,i m=1 a ni 2 i m x i2 x im 52 / 110

Tropical Eigenpairs Similarly, replace the classical operations on the right-hand side of the eigenpair equation by the tropical ones: 53 / 110

Tropical Eigenpairs Similarly, replace the classical operations on the right-hand side of the eigenpair equation by the tropical ones: x m 1 x (m 1) 1 λx [m 1] x m 1 1 = λ 2. λ x (m 1) 2 (H-eigenpair). xn m 1 xn (m 1) 54 / 110

Definition A tropical H-eigenpair for a tensor (a i1 i m ) R nm (x, λ) R n /R(1, 1,..., 1) R such that is a pair n i 2,...,i m=1 a ii2 i m x i2 x im = λ x m 1 i, i = 1, 2,..., n. We call x a tropical H-eigenvector and λ a tropical H-eigenvalue. Basically, Ax m 1 = λx [m 1] in tropical arithmetic. Similarly, a tropical E-eigenpair satisfies n i 2,...,i m=1 a ii2 i m x i2 x im = λ x i, i = 1, 2,..., n. 57 / 110

Example Take n = 2 and m = 3. Then a tropical H-eigenpair (x, λ) satisfies min{a 111 + 2x 1, a 112 + x 1 + x 2, a 121 + x 2 + x 1, a 122 + 2x 2 } = λ + 2x 1 min{a 211 + 2x 1, a 212 + x 1 + x 2, a 221 + x 2 + x 1, a 222 + 2x 2 } = λ + 2x 2. Notice that the tensor can be assumed to be symmetric in its last m 1 coordinates Example Take n = 3 and m = 3. Then a tropical H-eigenpair (x, λ) satisfies min{a 111 +2x 1,a 112 +x 1 +x 2,a 113 +x 1 +x 3,a 122 +2x 2,a 123 +x 2 +x 3,a 133 +2x 3 }=λ+2x 1 min{a 211 +2x 1,a 212 +x 1 +x 2,a 213 +x 1 +x 3,a 222 +2x 2,a 223 +x 2 +x 3,a 233 +2x 3 }=λ+2x 2 min{a 311 +2x 1,a 312 +x 1 +x 2,a 313 +x 1 +x 3,a 322 +2x 2,a 323 +x 2 +x 3,a 333 +2x 3 }=λ+2x 3. 61 / 110

Tropical E-Eigenpairs Initially, I focused on E-eigenpairs. Professor Sturmfels and I ran some experiments: 63 / 110

Tropical E-Eigenpairs Initially, I focused on E-eigenpairs. Professor Sturmfels and I ran some experiments: For randomly generated symmetric 3x3x3-tensors, we obtained the following distribution of tensors by number of tropical E-eigenpairs (from 0 eigenpairs to 7): [0, 80.1, 0.14, 19, 0, 0.12, 0.02, 0.58]% Conclusion: different (generic) tensors have a different number of E-eigenpairs and it is not obvious what the pattern of possible numbers of E-eigenpairs is. 66 / 110

Tropical H-Eigenpairs Theorem A tensor A R nm has a unique tropical H-eigenvalue λ(a) R. 68 / 110

Tropical H-Eigenpairs Theorem A tensor A R nm has a unique tropical H-eigenvalue λ(a) R. Proof idea: 69 / 110

Tropical H-Eigenpairs Theorem A tensor A R nm has a unique tropical H-eigenvalue λ(a) R. Proof idea: Existence: tropical analogue of proof of Perron-Frobenius Theorem. 70 / 110

Tropical H-Eigenpairs Theorem A tensor A R nm has a unique tropical H-eigenvalue λ(a) R. Proof idea: Existence: tropical analogue of proof of Perron-Frobenius Theorem. Proof of existence works for E-eigenpairs as well 71 / 110

Tropical H-Eigenpairs The H-eigenvalue is also a solution to an LP 73 / 110

Tropical H-Eigenpairs The H-eigenvalue is also a solution to an LP This LP simply requires λ to be a subeigenvalue: 74 / 110

Tropical H-Eigenpairs The H-eigenvalue is also a solution to an LP This LP simply requires λ to be a subeigenvalue: Theorem λ(a) = [ maximize λ subject to a i1 i 2 i m + x i2 +... + x im λ + (m 1)x i1 Dually, the H-eigenvalue of A is given by minimize (i 1,i 2,...,i m) [n] a m i1 i 2 i m y i1 i 2 i m subject to (i 1,i 2,...,i m) [n] y m i1 i 2 i m ((m 1)e i1 e i2... e im ) = 0 (i 1,i 2,...,i m) [n] y m i1 i 2 i m = 1 y i1 i 2 i m 0. ] 76 / 110

Interpreting the Dual We associate to a tensor A a weighted directed hypergraph H(A). The nodes are [n] = {1, 2,..., n} 77 / 110

Interpreting the Dual We associate to a tensor A a weighted directed hypergraph H(A). The nodes are [n] = {1, 2,..., n} The hyperedges are pairs (i 1, {i 2,..., i m }) 78 / 110

Interpreting the Dual We associate to a tensor A a weighted directed hypergraph H(A). The nodes are [n] = {1, 2,..., n} The hyperedges are pairs (i 1, {i 2,..., i m }) (we think of i 1 as having multiplicity (m 1)) 79 / 110

The dual problem is minimize (i 1,i 2,...,i m) [n] m a i 1 i 2 i m y i1 i 2 i m subject to (i 1,i 2,...,i m) [n] m y i 1 i 2 i m ((m 1)e i1 e i2... e im ) = 0 (i 1,i 2,...,i m) [n] m y i 1 i 2 i m = 1 y i1 i 2 i m 0. 82 / 110

The dual problem is minimize (i 1,i 2,...,i m) [n] m a i 1 i 2 i m y i1 i 2 i m subject to (i 1,i 2,...,i m) [n] m y i 1 i 2 i m ((m 1)e i1 e i2... e im ) = 0 (i 1,i 2,...,i m) [n] m y i 1 i 2 i m = 1 y i1 i 2 i m 0. We think of y i1 i 2 i m as the amout of flow through edge (i 1, {i 2,..., i m }) The term y i1 i 2 i m ((m 1)e i1 e i2... e im ) accounts for the amount of flow entering each of i 2,..., i m and leaving i 1 Condition 1 then states that we have a circulation (conservation of flow) 85 / 110

Interpreting the Dual Definition The circulation polytope H n,m is the polytope defined by the inequalities (i 1,i 2,...,i m) [n] m y i 1 i 2 i m ((m 1)e i1 e i2... e im ) = 0 (i 1,i 2,...,i m) [n] m y i 1 i 2 i m = 1 y i1 i 2 i m 0. The H-cycles are the vertices of H n,m. The H-eigenvalue is the minimum A-weighted H-cycle 90 / 110

H-cycles The H-cycles become more complicated for tensors compared with matrices 92 / 110

H-cycles The H-cycles become more complicated for tensors compared with matrices In particular, the vertices of the circulation polytope are no longer normalized characteristic functions of subsets of the edges Example The tensor whose entries are all zero except for a 132 = a 213 = a 322 = 1 is extremized on the vertex y 132 = 2 9, y 213 = 4 9, y 322 = 3 9, y ijk = 0 for remaining indices The edges here are 2 (1, {2, 3}), 4 (2, {1, 3}), 3 (3, {2, 2}) 94 / 110

Summary 95 / 110

Summary There exists a unique tropical H-eigenvalue 96 / 110

Summary There exists a unique tropical H-eigenvalue not at all the case for tropical E-eigenvalues 97 / 110

Summary There exists a unique tropical H-eigenvalue not at all the case for tropical E-eigenvalues it is equal to the minimum A-weighted H-cycle 98 / 110

Summary There exists a unique tropical H-eigenvalue not at all the case for tropical E-eigenvalues it is equal to the minimum A-weighted H-cycle can be found efficiently (LP) 99 / 110

Summary There exists a unique tropical H-eigenvalue not at all the case for tropical E-eigenvalues it is equal to the minimum A-weighted H-cycle can be found efficiently (LP) To a tensor there is an associated directed weighted hypergraph We have seen the circulation polytope, which is the polytope formed from the H-cycles its vertices are complicated for tensors 102 / 110

Future Directions 104 / 110

Future Directions Describe the tropical H-eigenvectors of a tensor 105 / 110

Future Directions Describe the tropical H-eigenvectors of a tensor Relate classical and tropical tensor spectral theories 106 / 110

Thank you Questions? 107 / 110

Existence and Uniqueness Theorem Let A R nm and assume that for each i = 1, 2,..., n, the sets S i := {{i (i) 2,..., i (i) m } : a ii (i) 2 i (i) m } are nonempty and mutually equal. Then A has a unique tropical H-eigenvalue λ(a) R. 108 / 110

Properties of H-cycle Polytope Theorem Let y R nm be a vertex of the H-cycle polytope H n,m. Then y has at most one nonzero entry of the form y ji2 i m for each j = 1, 2,..., n. There exist vertices with n nonzero entries. 109 / 110

Peter Butkovič. Max-linear systems: theory and algorithms. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2010. R. A. Cuninghame-Green. Describing industrial processes with interference and approximating their steady-state behaviour. OR, 13(1):pp. 95 100, 1962. Richard M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Math., 23(3):309 311, 1978. 110 / 110