张量的低秩逼近-Minru-Baippt

For A,B ∈ H, define the inner product and norm as
inner product norm
A rank-one tensor
unitary eigenvalue (U-eigenvalue) of T
Denote by Sym(d, n) all symmetric d-order n-dimensional tensors Let x ∈ Cn. Simply denote the rank-one tensor Define
2.张量特征值的计算
对称非负张量的最大H-特征值的计算： Perron-Frobenius 理论
1. 2. 3. 4. Ng, Qi, Zhou 2009， Chang, Pearson, Zhang 2011， L. Zhang, L. Qi 2012， Qi, Q. Yang, Y. Yang 2013
1. 张量的基本概念
张量的完备化
min rank(X ) s.t. X M
低秩张量M部分元素 M 被观察到,其中 是被观 察到的元数的指标集. 张量完备化是指:从所观察到的 部分元素来恢复逼近低秩张量M
1. 张量的基本概念
张量的特征值 H-特征值 Z（E）-特征值
m 1 m ' 1 Ax Bx m' n Bx 1 , C , x C
value, (,u) is a Z-eigenpair. [Qi 2011, Friedland2013, Zhang et al 2012]
[8] S. Friedland, Best rank one approximation of real symmetric tensors can be chosen symmetric, Frontiers of Mathematics in China, 8(2013), pp. 19-40. [9] X. Zhang, C. Ling and L. Qi, The best rank-1 approximation of a symmetric tensor and related spherical optimization problems, SIAM Journal on Matrix Analysis and Applications 33(2012), pp. 806-821.
[Cui, Dai, Nie 2014]
2. 张量特征值的计算
对称张量的US-特征值的计算：
1. Geometric measure of entanglement and U-eigenvalues of tensors, SIAM Journal on Matrix Analysis and Applications， [Ni，Qi，Bai 2014] 2. Complex Shifted Symmetric higher-order power method [Ni, Bai 2014]
is to minimizes the least-squares cost function
is said to be the best real rank-one approximation to tensor T. If T is a symmetric real tensor, is said to be
1. 张量的基本概念
张量的低秩逼近：用一个低秩的张量X近似表示张量A 最佳秩R逼近
min || A i ai (1) ai (2) ai ( N ) ||
i 1 R
最佳秩1逼近：R=1
min (1)
(1) (2) (N ) || A S X X X || 1 2 3 N (N)
3. 张量的秩1逼近和低秩逼近
张量的秩1逼近 •最佳实秩1逼近的计算方法： 交替方向法(ADM)、截断高阶奇异值分解(T-HOSVD)、 高阶幂法(HOPM) 和拟牛顿方法 等。----局部解，或稳定点 Nie, Wang[2013] ：半正定松弛方法 ----全局最优解 •最佳复秩1逼近的计算方法： Ni, Qi，Bai[2014] ：代数方程方法 ----全局最优解
结构张量，需要添加软件包（如：N-wayToolbox, CuBatch, PLS Toolbox, Tensor Toolbox）才能支持，其 中除PLS Toolbox外，都是免费软件。Tensor Toolbox是 支持稀疏张量。
• C++语言软件：HUJI Tensor Library (HTL)，FTensor,
Boost Multidimensional Array Library (Boost.MultiArray)
• FORTAN语言软件：The Multilinear Engine
5. 复张量的最佳秩1逼近和特征值
[A] Guyan Ni, Liqun Qi and Minru Bai, Geometric measure of entanglement and U-eigenvalues of tensors, SIAM Journal on Matrix Analysis and Applications 2014, 35(1): 73-87
[B] Guyan Ni, Minru Bai, Shifted Power Method for computing symmetric complex tensor US-eigenpairs, 2014, submitted.
Basic Definitions 1. A tensor S is called symmetric as its entries s_{i1· · · id } are invariant under any permutation of their indices. 2. A Z-eigenpair (, u) to a real symmetric tensor S is defined by
[10] G. Ni, L. Qi and M. Bai, Geometric measure of entanglement and U-eigenvalues of tensors, to appear in SIAM Journal on Matrix Analysis and Applications
对称张量的最大Z-特征值的计算：
1. The sequential SDPs method [Hu, Huang, Qi 2013]
2. Sequential subspace projection method[Hao, Cui, Dai. 2014]
3. Shifted symmetric higher-order power method [Kolda,Mayo 2011] 4. Jacobian semidefinite relaxations 计算对称张量所有实的B-特征值
Axm1 x[ m1] .
2005, Qi
Axm x[ m1]
B-特征值 US-特征值
2014，Cui, Dai, Nie
S * x m1 x* *m1 x Sx x*T x 1, C, x C n
2014，Ni, Qi, Bai
Tucker逼近
S,X
,, X
s.t. S R R1 R2 RN , X (i ) R ni Ri , X (i )T X (i ) I Ri , i 1, , N
(2) (N ) ( xi1i2 iN gr1r2 rN ai(1) a a iN rN ) 1r 1 i2 r2 r1 1 r2 1 rN 1 R1 R2 RN
4. The best rank-one tensor approximation problems Assume that T a d-order real tensor. Denote a rank-one tensor
.
Then the rank-one approximation problem
The rank-one tensor
the best real symmetric rank-one approximation.
Basic results
• Friedland [2013] and Zhang et al [2012] showed that the best real rank one approximation to a real symmetric tensor,
1阶张量：向量 2阶张量：矩阵 A=(aij) 3阶张量：长方体 A=(aijk)
1. 张量的基本概念
张量的秩
秩1张量：
X a(1) a(2) a( N ) , 其中 xi1i2iN ai(11) ai(22) ai(NN )
秩1矩阵：A=a bT = (aibj)
张量的秩： 1927年 Hitchcock
We call a number ∈ C a unitary symmetric eigenvalue (USeigenvalue) of S if and a nonzero vector
张量的低秩逼近
白敏茹
湖南大学数学与计量经济学院
ቤተ መጻሕፍቲ ባይዱ
2014-11-15
目 录
1. 张量的基本概念
2. 张量特征值的计算
3. 张量秩1逼近和低秩逼近
4. 张量计算软件
5. 复张量的最佳秩1逼近和特征值
1. 张量的基本概念
张量：多维数组 X [ xi i i ] Rn n n
1 2 12 N N
2005, Qi
uTu
3. An eigenpair (, u) to a real symmetric tensor S is defined by