Perron-Frobenius theorem for nonnegative multilinear forms and extensions
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1 Perron-Frobenius theorem for nonnegative multilinear forms and extensions Shmuel Friedland Univ. Illinois at Chicago ensors 18 December, 2010, Hong-Kong
2 Overview 1 Perron-Frobenius theorem for irreducible nonnegative matrices 2 Irreducibility and weak irreducibility for tensors 3 Perron-Frobenius for irreducible tensors 4 Rank one approximation of tensors 5 Nonnegative multilinear forms 6 Monotone homogeneous maps 7 Polynomial homogeneous maps 8 Collatz-Wielandt 9 Power iterations 10 Numerical counterexamples
3 Nonnegative irreducible and primitive matrices A = [a ij ] R n n + induces digraph DG(A) = DG = (V, E) V = [n] := {1,..., n}, E [n] [n], (i, j) E a ij > 0 DG strongly connected, SC, if for each pair i j there exists a dipath from i to j Claim: DG SC iff for each I [n] j [n] \ I s.t. (i, j) E A -primitive if A N > 0 for some N > 0 A N (R n + \ {0}) int R n + A primitive A irreducible and g.c.d of all cycles in DG(A) is one
4 Perron-Frobenius theorem PF: A R n + irreducible. Then 0 < ρ(a) spec (A) algebraically simple x, y > 0 Ax = ρ(a)x, A y = ρ(a)y. A R n n + primitive iff in addition to above λ < ρ(a) for λ spec (A) \ {ρ(a)} Collatz-Wielandt: ρ(a) = min x>0 max i [n] (Ax) i x i = max x>0 min i [n] (Ax) i x i
5 SVD A R m n, σ 1 (A)... 0 singular values Ay i = σ i (A)x i, A x i = σ i (A)y i ±σ i (A), i = 1,... are critical values of f (x, y) = x Ay restricted to x 2 = y 2 = 1 SVD[ of A closely related ] to spectral theory 0m m A B =, λ(b) = λ(b) A 0 n n positive singular values are the positive eigenvalues of B σ 1 (A) = max x 2 = y 2 =1 y Ax
6 SVD of nonnegative matrices Perron-Frobenius for A = [a ij ] R m n + : u R m +, v R n +, u u = v v = 1 Av = σ 1 (A)u, A u = σ 1 (A)v σ 1 (A) = max x R m,y R n, x 2 = y 2 =1 x Ay = u Av. G(A) := DG(B) = G(B) = (V 1 V 2, E) bipartite graph on V 1 = [m], V 2 = [n], (i, j) E a ij > 0. If G(A) connected. Then u, v unique, σ 2 (A) < σ 1 (A), ( as B-irreducible).
7 Irreducibility and weak irreducibility of nonnegative tensors F := [f i1,...,i d ] d i=1 Rm i = R m 1 m d is called d-tensor, (d 3) T 0 if T R m 1 m d + G(F) = (V, E(F)), V = d j=1 V j, d-partite graph V j = [m j ], j [d]. (i k, i l ) V k V l, k l is in E(F) iff f i1,i 2,...,i d > 0 for some d 2 indices {i 1,..., i d }\{i k, i l }. F weakly irreducible if G(F) is connected. F irreducible: for each I V, J := V \I there exists k [d], i k I V k and i j J V j for each j [d]\{k} such that f i1,...,i d > 0. Claim: irreducible implies weak irreducible
8 Perron-Frobenius theorem for nonnegative tensors I T = [t i1,...,i d ] d i=1 Cn maps C n to itself T (x) i = i 2,...,i d [n] t i,i 2,...,i d x i2... x id, i [n] T has eigenvector x = (x 1,..., x n ) C n with eigenvalue λ: T (x) i = λx d 1 i for all i [n] Assume: T 0, T (R n + \ {0}) R n + \ {0} 1 T 1 : Π n Π n, x 1 (T (x)) 1 d 1 n i=1 T (x) d 1 i Brouwer fixed point: x 0 eigenvector with λ > 0 eigenvalue Problem When there is a unique positive eigenvector with maximal eigenvalue?
9 Perron-Frobenius theorem for nonnegative tensors II Theorem Chang-Pearson-Zhang 2009 [2] Assume T ( d i=1 Rn ) + is irreducible. Then there exists a unique nonnegative eigenvector which is positive with the corresponding maximum eigenvalue λ. Furthermore the Collatz-Wielandt characterization holds λ = min x>0 max i [n] (T (x)) i x d 1 i = max x>0 min i [n] (T (x)) i x d 1 i
10 Rank one approximations for 3-tensors R m n l IPS: A, B = m,n,l i=j=k a i,j,kb i,j,k, T 2 = T, T x y z, u v w = (u x)(v y)(w z)
11 Rank one approximations for 3-tensors R m n l IPS: A, B = m,n,l i=j=k a i,j,kb i,j,k, T 2 = T, T x y z, u v w = (u x)(v y)(w z) X subspace of R m n l, X 1,..., X d an orthonormal basis of X P X (T ) = d i=1 T, X i X i, P X (T ) 2 2 = d i=1 T, X i 2 T 2 2 = P X(T ) T P X(T ) 2 2 Best rank one approximation of T :
12 Rank one approximations for 3-tensors R m n l IPS: A, B = m,n,l i=j=k a i,j,kb i,j,k, T 2 = T, T x y z, u v w = (u x)(v y)(w z) X subspace of R m n l, X 1,..., X d an orthonormal basis of X P X (T ) = d i=1 T, X i X i, P X (T ) 2 2 = d i=1 T, X i 2 T 2 2 = P X(T ) T P X(T ) 2 2 Best rank one approximation of T : min x,y,z T x y z 2 = min x 2 = y 2 = z 2 =1,a T a x y z 2 Equivalent: max m,n,l x 2 = y 2 = z 2 =1 i=j=k t i,j,kx i y j z k Lagrange multipliers: T y z := j=k=1 t i,j,ky j z k = λx T x z = λy, T x y = λz λ singular value, x, y, z singular vectors
13 Nonnegative multilinear forms Associate with T = [t i1,...,i d ] R m 1... m d + a multilinear form f (x 1,... x d ) : R m 1... m d R f (x 1,..., x d ) = i j [m j ],j [d] t i 1,...,i d x i1,1... x id,d, x i = (x 1,i,..., x mi,i R m i For u R m, p (0, ] let u p := ( m i=1 u i p ) 1 p and S m 1 p,+ := {0 u Rm, u p = 1} For p 1,..., p d (1, ) critical point (ξ 1,..., ξ d ) S m 1 1 p 1,+ of f S m 1 1 p 1,+... Sm d 1 p d,+ satisfies Lim [1]: ti1,...,i d x i1,1... x ij 1,j 1x ij+1,j+1... x id,d = λx p j 1 i j,j, i j [m j ], x j S p j 1 m j,+, j [d]... Sm d 1 p d,+
14 Perron-Frobenius theorem for nonnegative multilinear forms Theorem- Friedland-Gauber-Han [3] f : R m 1... R m d R, a nonnegative multilinear form, T weakly irreducible and p j d for j [d]. Then f has unique positive critical point on S m S m d 1 +. If F is irreducible then f has a unique nonnegative critical point which is necessarily positive
15 Eigenvectors of homogeneous monotone maps on R n + Hilbert metric on PR n >0 : for x = (x 1,..., x n ), y = (y 1,..., y n ) > 0. Then dist(x, y) = max i [n] log y i x i min i [n] log y i x i. F = (F 1,..., F n ) : R n >0 Rn >0 homogeneous: F(tx) = tf(x) for t > 0, x > 0, and monotone F(y) F(x) if y x > 0. F induces ˆF : PR n >0 PRn >0 F nonexpansive with respect to Hilbert metric dist(f(x), F(y)) dist(x, y). α max x y β min x α max F(x) = F(α max x) F(y) F(β min x) = β min F(x) dist(f(x), F(y)) log β min α max = dist(x, y) x > 0 eigenvector of F if F(x) = λf(x). So x PR n + fixed point of F PR n +.
16 Existence of positive eigenvectors of F 1. If F contraction: dist(f(x), F(y)) K dist(x, y) for K < 1, then F has unique fixed point in PR n + and power iterations converge to the fixed point 2. Use Brouwer fixed and irreducibility to deduce existence of positive eigenvector 3. [4, Theorem 2]: for u (0, ), J [n] let u J = (u 1,..., u n ) > 0: u i = u if i J and u i = 1 if i J. F i (u J ) nondecreasing in u. di-graph G(F) [n] [n]: (i, j) G(F) iff lim u F i (u {j} ) =. Thm: F : R n >0 Rn >0 homogeneous and monotone. If G(F) strongly connected then F has positive eigenvector
17 Uniqueness and convergence of power method for F Thm 2.5, Nussbaum 88: F : R n >0 Rn >0 homogeneous and monotone. Assume; u > 0 eigenvector F with the eigenvalue λ > 0, F is C 1 in some open neighborhood of u, A = DF(u) R+ n n ρ(a)(= λ) a simple root of det(xi A). Then u is a unique eigenvector of F in R n >0. Cor 2.5, Nus88: In the above theorem assume A = DF(u) is primitive. Let ψ 0, ψ u = 1. Define G : R n >0 Rn >0 G(x) = 1 ψ F(x) F(x) Then lim m G m (x) = u for each x R n >0.
18 Outline of the uniqueness of pos. crit. point of f Define: F : R m + R n + R l + R m + R n + R l +: ( F((x, y, z)) i,1 = x p 3 p ( F((x, y, z)) j,2 = y p 3 p ( F((x, y, z)) k,3 = z p 3 p ) 1 n,l j=k=1 t p 1 i,j,ky j z k, i = 1,..., m ) 1 m,l i=k=1 t p 1 i,j,kx i z k, j = 1,..., n ) 1 m,n i=j=1 t p 1 i,j,kx i y j, k = 1,..., l Assume n,l j=k=1 t i,j,k > 0, i = 1,..., m, m,l i=k=1 t i,j,k > 0, j = 1,..., n, m,n i=j=1 t i,j,k > 0, k = 1,..., l F 1-homogeneous monotone, maps open positive cone R m + R n + R l + to itself. T = [t i,j,k ] induces tri-partite graph on m, n, l : i m connected to j n and k l iff t i,j,k > 0, sim. for j, k If tri-partite graph is connected then F has unique positive eigenvector If F completely irreducible, i.e. F N maps nonzero nonnegative vectors to positive, nonnegative eigenvector is unique and positive
19 Perron-Frobenius theorem for nonnegative polynomial maps P = (P 1,..., P n ) : R n R n polynomial map deg P i = d i 1 with nonnegative coefficients. Let δ i d i, i [n]. Consider the system P i (x) = λx δ i i, i [n], x 0 Assume P weakly irreducible. Then for each a, p > 0!x > 0, depending on a, p, satisfying above equation and x p = a. If P irreducible then above system has a unique solution, depending on a, p satisfying x p = a, with all coordinates positive
20 Collatz-Wielandt Collatz-Wielandt [1, Lemma 2.8]: P = (P 1,..., P n ) : R n R n, P i homogeneous polynomial of degree d 1 with nonnegative coefficients. Assume P weakly irreducible. Then there exists unique scalar λ, u with P i (u) = λui d, i [n] which satisfies P λ = inf x interior R n + max i (x) i [n] = x d sup x R n + \{0} min i [n] x i 0 P i (x) x d i i
21 Geometric convergence of power algorithm P weakly primitive if the di-graph G(P) is strongly connected and if the gcd of the lengths of its circuits is equal to one. Cor. 2.5, Nussbaum 88 yields Thm: Let P and d be above and assume that P is weakly primitive. Then x (k+1) i = (ψ F(x (k) )) 1 F i (x (k) ), k = 1,..., converges to the unique vector u interior R n + satisfying P i (u) = λui d, i [n], where ψ u = 1
22 Numerical counterexamples
23 Numerical counterexamples F := [f i,j,k ] R : f 1,1,1 = f 2,2,2 = a > 0 otherwise, f i,j,k = b > 0. f (x, y, z) = b(x 1 + x 2 )(y 1 + y 2 )(z 1 + z 2 ) + (a b)(x 1 y 1 z 1 + x 2 y 2 z 2 ).
24 Numerical counterexamples F := [f i,j,k ] R : f 1,1,1 = f 2,2,2 = a > 0 otherwise, f i,j,k = b > 0. f (x, y, z) = b(x 1 + x 2 )(y 1 + y 2 )(z 1 + z 2 ) + (a b)(x 1 y 1 z 1 + x 2 y 2 z 2 ). For p 1 = p 2 = p 3 = p > 1 positive singular vectors: x = y = z = (0.5 1/p, 0.5 1/p ).
25 Numerical counterexamples F := [f i,j,k ] R : f 1,1,1 = f 2,2,2 = a > 0 otherwise, f i,j,k = b > 0. f (x, y, z) = b(x 1 + x 2 )(y 1 + y 2 )(z 1 + z 2 ) + (a b)(x 1 y 1 z 1 + x 2 y 2 z 2 ). For p 1 = p 2 = p 3 = p > 1 positive singular vectors: x = y = z = (0.5 1/p, 0.5 1/p ). For a = 1.2, b = 0.2 and p = 2 additional positive singular vectors: x = y = z (0.9342, ), x = y = z (0.3568, ).
26 Numerical counterexamples F := [f i,j,k ] R : f 1,1,1 = f 2,2,2 = a > 0 otherwise, f i,j,k = b > 0. f (x, y, z) = b(x 1 + x 2 )(y 1 + y 2 )(z 1 + z 2 ) + (a b)(x 1 y 1 z 1 + x 2 y 2 z 2 ). For p 1 = p 2 = p 3 = p > 1 positive singular vectors: x = y = z = (0.5 1/p, 0.5 1/p ). For a = 1.2, b = 0.2 and p = 2 additional positive singular vectors: x = y = z (0.9342, ), x = y = z (0.3568, ). For a = 1.001, b = and p = 2.99 additional positive singular vectors: x = y = z (0.9667, ), x = y = z (0.4570, )
27 References I M. Akian, S. Gaubert, and A. Guterman. Tropical polyhedra are equivalent to mean payoff games, Eprint arxiv: , K.C. Chang, K. Pearson, and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6, (2008), S. Friedland, S. Gauber and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms, arxiv: S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc. 356 (2004),
28 References II L.H. Lim, Singular values and eigenvalues of tensors: a variational approach, CAMSAP 05, 1 (2005), M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl. 31 (2009), R. Nussbaum, Hilbert s projective metric and iterated nonlinear maps, Memoirs Amer. Math. Soc., 1988, vol. 75. L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005),
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