The Miracles of Tropical Spectral Theory

Transcription

1 The Miracles of Tropical Spectral Theory Emmanuel Tsukerman University of California, Berkeley July 31, / 110

2 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

3 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

4 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 Credit goes to Bernd Sturmfels for suggesting this avenue of research 4 / 110

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

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

7 Basic Tropical Geometry 7 / 110

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

9 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

10 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) = = 7 8 = 7 10 / 110

11 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) = = 7 8 = 7 Associative, distributive, commutative 11 / 110

12 Tropical Arithmetic There is also an additive identity 12 / 110

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

14 Tropical Arithmetic Matrix times vector multiplication: 14 / 110

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

16 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

17 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

18 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

19 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 ( ) λ + a11 λ + a = 12 λ + a 21 λ + a 22 ) 19 / 110

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

21 Tropical Spectral Theory of Matrices 21 / 110

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

23 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

24 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

25 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

26 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

27 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 27 / 110

28 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: a i0 i 1 + a i1 i a ik 1 i k. 28 / 110

29 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 a ik 1 i k. a i0 i 1 + a i1 i a ik 1 i k k is called the normalized length of the path. 29 / 110

30 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 a ik 1 i k. a i0 i 1 + a i1 i a ik 1 i k k is called the normalized length of the path. Directed cycles are paths for which i 0 = i k. 30 / 110

31 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). 31 / 110

32 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

33 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) However, there is an efficient algorithm based on linear programming given by Karp [Kar78]: 33 / 110

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

35 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 ] 35 / 110

36 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

37 Classical and Tropical Eigenvalues 37 / 110

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

39 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

40 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

41 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. lim ρ(x (k) ) 1 k = λ(x ). k 41 / 110

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

43 Classical Tensor Eigenpairs 43 / 110

44 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

45 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

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

47 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

48 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

49 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

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

51 Tropical Spectral Theory of Tensors 51 / 110

52 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

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

54 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

55 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) and λx λ x (E-eigenpair) 55 / 110

56 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. 56 / 110

57 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

58 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. From now on, I may use E-eigenpairs and H-eigenpairs to shorten for tropical E-eigenpairs and H-eigenpairs when there is no danger of confusion 58 / 110

59 Example Take n = 2 and m = 3. Then a tropical H-eigenpair (x, λ) satisfies min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 2 } = λ + 2x 1 min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 2 } = λ + 2x / 110

60 Example Take n = 2 and m = 3. Then a tropical H-eigenpair (x, λ) satisfies min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 2 } = λ + 2x 1 min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 2 } = λ + 2x 2. Notice that the tensor can be assumed to be symmetric in its last m 1 coordinates 60 / 110

61 Example Take n = 2 and m = 3. Then a tropical H-eigenpair (x, λ) satisfies min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 2 } = λ + 2x 1 min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 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 x 1,a 112 +x 1 +x 2,a 113 +x 1 +x 3,a x 2,a 123 +x 2 +x 3,a x 3 }=λ+2x 1 min{a x 1,a 212 +x 1 +x 2,a 213 +x 1 +x 3,a x 2,a 223 +x 2 +x 3,a x 3 }=λ+2x 2 min{a x 1,a 312 +x 1 +x 2,a 313 +x 1 +x 3,a x 2,a 323 +x 2 +x 3,a x 3 }=λ+2x / 110

62 Example Take n = 2 and m = 3. Then a tropical H-eigenpair (x, λ) satisfies min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 2 } = λ + 2x 1 min{a x 1, a x 1 + x 2, a x 2 + x 1, a x 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 x 1,a 112 +x 1 +x 2,a 113 +x 1 +x 3,a x 2,a 123 +x 2 +x 3,a x 3 }=λ+2x 1 min{a x 1,a 212 +x 1 +x 2,a 213 +x 1 +x 3,a x 2,a 223 +x 2 +x 3,a x 3 }=λ+2x 2 min{a x 1,a 312 +x 1 +x 2,a 313 +x 1 +x 3,a x 2,a 323 +x 2 +x 3,a x 3 }=λ+2x 3. For E-eigenpairs, we would replace 2x 1, 2x 2, 2x 3 on the right with x 1, x 2, x / 110

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

64 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): 64 / 110

65 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]% 65 / 110

66 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

67 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. In contrast, we have the following result for H-eigenpairs 67 / 110

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

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

70 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

71 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

72 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 Uniqueness: Gordan s Theorem (a theorem of the alternative) 72 / 110

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

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

75 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 i x im λ + (m 1)x i1 ] 75 / 110

76 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 i 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

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

78 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

79 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

80 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)) 80 / 110

81 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)) Each hyperedge (i 1, {i 2,..., i m }) receives a weight a i1 i 2 i m. 81 / 110

82 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 / 110

83 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 }) 83 / 110

84 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 84 / 110

85 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

86 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) The a s (tensor entries) give a weight to each edge 86 / 110

87 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) The a s (tensor entries) give a weight to each edge Interpretation: we seek to find a nontrivial circulation with least A-weight 87 / 110

88 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 / 110

89 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. 89 / 110

90 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

91 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 For matrices, the circulation polytope is called the normalized cycle polytope 91 / 110

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

93 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 93 / 110

94 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

95 Summary 95 / 110

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

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

98 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

99 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

100 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 100 / 110

101 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 101 / 110

102 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

103 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 tensors correspond to linear functionals on this polytope, and extrema correspond to tropical H-eigenvalues 103 / 110

104 Future Directions 104 / 110

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

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

107 Thank you Questions? 107 / 110

108 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

109 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

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