10384 19020101152519 UDC Rayleigh Quasi-Rayleigh Method for computing eigenvalues of symmetric tensors 2 0 1 3 2 0 1 3 2 0 1 3 2013
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[4], [27].,. [6] E- ; [7], Z-. [15]. Ramara G. kolda [1, 2], (1) (SS-HOPM),. max subject to x 2 = 1 (1) x R n. Rayleigh. (S-HOPM). m,, m,. (SS-HOPM),., SS-HOPM. SS- HOPM α, SS-HOPM α µ k = Ax m k x k x k. µ k A Rayleigh, 3 Rayleigh (QRS-HOPM)., Rayleigh (QRS-HOPM). QRS-HOPM SS-HOPM, : QRS-HOPM k qrs hopm SS-HOPM k ss hopm, Rayleigh (QRS-HOPM) (SS-HOPM).,,, Rayleigh. I
Abstract Work on eigenvalues and eigenvectors for tensor has been applied in blind source separation[4], magnetic resonance imaging[27], molecular conformation, and more. Recent years, eigenvalue problem for tensors attracts researchers attention in application domain. Work by Qi[6,7] discussed the characteristic polynomial for real tensors, the definition and properties of E-eigenvalue and Z-eigenvalue for tensors. Also, professor zhang[15] discussed the rank for tensors. Recent work by Tamaka G.Kalda and Jackson R. Mayo [2] discussed the shifted symmetric higher- order power method(ss-hopm) of the problem(2), which show that converge to a tensor eigenvalue. max subject to x 2 = 1 (2) x R n Continuing along this vein, this paper considers the case that symmetric higherorder quasi-rayleigh method(qrs-hopm), which we show is guaranteed to converge to a tensor eigenvalue. The chapter one, we introduce some background knowledge of tensers. The chapter two and three, to introduce S-HOPM, SS-HOPM and their numerical experiments analysis. Where S-HOPM converge to the largest eigenvalue when the order of tenser is even, otherwise S-HOPM is not convergence to a eigenvalue of tensor. But SS-HOPM is uniform convergence to the largest eigenvalue of the tensor. The chapter four, based on the mathematical calculation, we substitute µ k = Axm x x for α that in the SS-HOPM(Tamara G.Kolda [2]) at each iteration steps. We call this improved algorithm Quasi-Rayleigh Method for computing symmetric higher-order tensor eigenvalue(qrs-hopm). To apply the fixed point theorem[2,th4.4], we can characterize which eigenvalue can be found by QRS-HOPM. Finally we present this iterative method for finding the eigenvalues of tensors, and comparing its convergence rate k qrs hopm with the convergence rate k ss hopm of the SS-HPOM. Then we get the conclusion that is k qrs hopm always less than k ss hopm. Key Words: Eigenvalue, symmetric tensor, SS-HOPM, QRS-HOPM. III
... I Abstract... III... 1 1.1................................................ 1 1.2......................................................... 3 1.3.................................................. 4 1.4 Rayleigh..................................................... 5 S HOPM... 7 2.1................................................ 7 2.2 S-HOPM................................. 9 SS HOPM... 11 3.1................................................ 11 3.2 SS-HOPM........................... 15 Rayleigh (QRS-HOPM)... 17 4.1 Rayleigh (QRS-HOPM)................... 17 4.2 Rayleigh (QRS-HOPM)..................... 20 4.3................................................................... 27 V
... 35... 38 VI
CONTENTS Abstract in Chinese.....................................................I Abstract in English................................................... III Chapter 1 Introduction of tensor................................... 1 1.1 Eigenvalue of a tensor............................................ 1 1.2 Introduction of convex function.................................. 3 1.3 Constrained function optimization............................... 4 1.4 Rayleigh of a tensor.............................................. 5 Chapter 2 Symmetric higher-order power method.............. 7 2.1 Properties of symmetric higher-order power tensor............. 7 2.2 Numerical analysis of S-HOPM.................................. 9 Chapter 3 Shifted symmetric higher-order power method.... 11 3.1 Properties of shifted symmetric higher-order power tensor..... 11 3.2 Numerical analysis of S-HOPM.................................15 Chapter 4 symmetric higher-order quasi-rayleigh method(qrs-hopm).................................................17 4.1 Convergence analysis of QRS-HOPM.......................... 17 4.2 Numerical analysis of QRS-HOPM............................. 20 4.3 Conclusion...................................................... 27 References............................................................... 35 Acknowledgements.................................................... 38 VII
Gauss, Riemann Christoffel., Ricci, Levi-Civita. 1916, Einstein,.,. [4], [27]. Newton [5] HOPM[2] S-HOPM SS-HOPM[1, 2]. 1.1 R [m,n] m n, R [3,2] = R 2 2 2., ( [18] Z [8] l 2 ). 1. 1([6, 7]) A R [m,n], x R n, n n (Ax m 1 ) i1... a i1 i 2...i m x i2... x im for i 1 = 1,, n (1.1) i 2 =1 x R n, i m=1 Ax m 1 = λx for x x = 1 (1.2) λ R A, x λ, (λ, x) A. [6,7,8] (x, λ) (1. 3) Karush Kuhn Tucker(KKT). max subject to x x = 1 (1.3) x R n Ax m n i 1 =1 n a i1 i m x i1 x im i m=1 (1.3) rank-1 [25]. 1
1.1,, [2], Γ Σ R n, Γ = {x R n : x 1} Σ = {x R n : x = 1}, Π m {1,, m} x = {y R n : x y}, x y x y. 1.2([1,6]) (1.4), A R [m,n]. a ip(1) ip(m) = a i1 i m for all i 1 i m 1,, n (1.4) P = Π m 1. 3([2,8]) A R [m,n], x R n, r [0, m 1], A x (m r) Ax m r R [r,n] (Ax m r ) i1 i r = i r+1 i m a i1 i m x ir+1 x im for all i 1,... i m 1,, n,.,,., 1.4. 1.4([2]) (1.5), E R [m,n]. Ex [m 1] = x for all x Σ (1.5) x Σ, x R n, Ex [m 1] = x [m 1] E( x x )[m 1] = x [m 1] x = x x [m 2] x x / Σ, x [m 2] x x Ex [m 1] x, x Σ, E.,,., [2,6,7]. N,, 2 3 4 3 6 4 2 12 3 8. N 2
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