张量的低秩逼近
白敏茹
湖南大学数学与计量经济学院
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
12
1阶张量：向量
2阶张量：矩阵 A=(aij)
3阶张量：长方体 A=(aijk)
n1n2 
]

R
iN
nN
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 xi1i2iN  ai(11) ai(22) ai(NN )
秩1矩阵：A=a bT = (aibj)
张量的秩： 1927年 Hitchcock
R
rank ( X )  min{R | X   ai (1) ai (2)
i 1
ai ( N ) }
NP-Hard
n-rank rankn ( X )  (rank ( X (1) ), , rank ( X ( N ) )) 可计算
其中
X (k ) 表示 张量X的mode-k mode
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
R
最佳秩R逼近
min || A   i ai (1) ai (2)
i 1
ai ( N ) ||
最佳秩1逼近：R=1
Tucker逼近
S,X
min
(1)
, ,X
(1)
(2)
||
A

S

X

X
3
1
2
(N)
 N X ( N ) ||
s.t. S  R R1 R2  RN ,
X (i )  R ni  Ri , X (i )T X (i )  I Ri , i  1,
R1
( xi1i2
iN
R2
 
r1 1 r2 1
RN
 gr1r2
rN 1
(1) (2)
a
rN i1r1 ai2 r2
ai(NNrN) )
,N
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
Axm1   x[ m1] .
2005, Qi
Z（E）-特征值
B-特征值
US-特征值
Axm  x[ m1]
 Axm 1  Bxm '1
 m'
 Bx  1,   C , x  C n
S * x m1   x*
 *m1
 x
Sx
 x*T x  1,   C, x  C n

2014，Cui, Dai, Nie
2014，Ni, Qi, Bai
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
 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
[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]
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
3. 张量的秩1逼近和低秩逼近
 张量的低秩逼近
•最佳秩R逼近的计算方法：
交替最小平方法(ALS)
•最佳Tucker逼近的计算方法：
高阶奇异值（HOSVD），TUCKALS3，t-SVD
4. 张量计算软件
• Matlab, Mathematica, Maple都支持张量计算
• Matlab仅支持简单运算，而对于更一般的运算以及稀疏和
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
2005, Qi
uTu
3. An eigenpair (, u) to a real symmetric tensor S is defined by
2011, Kolda and Mayo
u*Tu
[7] T.G. Kolda and J.R. Mayo, Shifted power method for computing tensor eigenpairs,
SIAM Journal on Matrix Analysis and Applications, 32(2011), pp. 1095-1124.
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
is to minimizes the least-squares cost function
The rank-one tensor
is said to be the best real
rank-one approximation to tensor T.
If T is a symmetric real tensor,
is said to be
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,
which in principle can be nonsymmetric, can be chosen
symmetric.
•  ud is the best real rank-one approximation of T if and
only if  is a Z-eigenvalue of T with the largest absolute
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.
complex tensors and unitary eigenvalues
A d-order complex tensor will be denoted by
inner product
norm
The superscript * denotes the complex conjugate.
The superscript T denotes transposition.
[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
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
We call a number  ∈ C a unitary symmetric eigenvalue (USeigenvalue) of S if  and a nonzero vector
The largest |λ| is the entanglement eigenvalue. The corresponding
rank-one tensor ⊗di =1x is the closest symmetric separable state.
Theorem 1. Assume that complex d-order tensors
Then
b) all U-eigenvalues are real numbers;
c) the US-eigenpair (, x) to a symmetric d-order complex
tensor S can also be defined by the following equation system
or
(1)
3.1. US-eigenpairs of symmetric tensors
Case d=2:
Theorem 3. (Takagi’s factorization) Let A ∈ Cn×n be a complex
symmetric tensor. Then there exists a unitary matrix U ∈ Cn×n
such that
Theorem 4. Let A ∈ Cn×n be a complex symmetric tensor. Let U ∈
Cn×n be a unitary matrix such that
Let ei = (0, · · · , 0, 1, 0, · · · , 0)T , i = 1, · · · , n.
and
Then both
are US-eigenpairs of A.
The number of distinct US-eigenvalues is at most 2n.
Theorem 5. If  1 = · · · =  k >  k+1, 1 ≤ k ≤ n, then the set of all
US-eigenvectors with respect to  1 is
the set of all US-eigenvectors with respect to −λ1 is
3.2. US-eigenpairs of symmetric tensors
Case d  3
 The problem of finding eigenpairs is equivalent to solving
a polynomial system
[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.
3.2. US-eigenpairs of symmetric tensors
Case d  3
Theorem 2. Assume that a complex d-order n-dimension symmetric
tensor S ∈ Sym(d, n). Then
a) if d ≥ 3, d is an odd integer, and   0, then the system (1) is
equivalent to
(2)
and the number of US-eigenpairs of (1) is the double of the
number of solutions of (2);
b) if d ≥ 3, d is an even integer, and   0, then the system (1) is
equivalent to
(3)
and the number of US-eigenpairs of (1) is equal to the number of
solutions of (3).
3.2. US-eigenpairs of symmetric tensors
Case d  3
Theorem 6. Let d ≥ 3, n ≥ 2 be integers, S ∈ Sym(d, n). If (2) has
finitely many solutions, then
a) if d is odd , the number of non-zero solutions of (2) is at most
b) if d is even, the number of non-zero solutions of (3) is at most
c) S has at most
distinct nonzero US-eigenvalues;
d) for nonzero US-eigenvalues, all the US-eigenpairs of S are as follows
where x is a solution of (2).
Note. 1. Let S be the symmetric 2 × 2 × 2 × 2 tensor whose nonzero entries are
S1111 = 2, S1112 = −1, S1122 = −1, S1222 = −2, S2222 = 1.
The number of non-zero solutions of the equation system (2) is 40
which shows that the bound is tight.
Note. 2. Cartwright and Sturmfels (2013) showed that every symmetric
tensor has finite E-eigenvalues. At the same time, they indicated that the
magnitudes of the eigenvalues with ||x|| = 1 may still be an infinite set
(See Example 5.8 of [Cartwright and Sturmfels (2013) ]), which implies
that the system S x d−1 = x has infinite non-zero solutions, where S is a
symmetric 3 × 3 × 3 tensor whose non-zero entries are
S111 = 2, S122 = S212 = S221 = S133 = S313 = S331 = 1.
[11] D. Cartwright and B. Sturmfels, The number of eigenvalues of a tensor, Linear
Algebra and its Applications, 438(2013), pp. 942-952
Note. 3. Let S be the symmetric 3×3×3 tensor as in Note 2. Then
x=
for all 0 < a < 1 are nonzero solutions of Sx d−1 = x*. It implies that (2) may have infinite
non-zero solutions.
4. Best symmetric rank-one approximation
of symmetric tensors
Theorem 7. Let S be a symmetric complex tensor. Let  be a USeigenvalue of S. Then
a) −  is also a US-eigenvalue of S ;
b) G(S) =  max.
Case d=2
Theorem 8. Let A ∈ Cn×n be a complex symmetric matrix.
Then for all x ∈ UEV (A,  1) ∪ UEV (A,−  1) and  ∈ C
with || = 1, ( x) ⊗ ( x) are best symmetric rank-one
approximation of A.
Case d ≥ 3
The best symmetric rank-one approximation problem is to find a
unit-norm vector x ∈ Cn, such that
By Theorem 7, introducing the US-eigenvalue method, Q1 is
equivalent to the following problem
Theorem 9. Let S ∈ Sym(d, n). Then
a) the best symmetric rank-one approximation problem is
equivalent to the following optimization problem
rank-one
approximation of S for each
 The problem of finding eigenpairs is equivalent to solving
a polynomial system
Let x = y + z−1, y, z ∈ Rn.
following problem
Then Q3 is equivalent to the
Example 1. Assume that S is a real symmetric tensor with d = 3 and n
= 2. Then Q4 is equivalent to the following optimization problem
Table1. US-eigenpairs of S with S111 = 2, S112 = 1, S122 = −1, S222 = 1.
The best real rank-one approximation is also the best
complex rank-one approximation.
The absolute-value largest of Z-eigenvalues is not
its largest US-eigenvalue.
By observing numerical examples, we find the following results:
 The best real rank-one approximation is sometimes also
the best complex rank-one approximation even if the
tensor is not a symmetric nonnegative real tensor, see
Table 1.
 The absolute-value largest of Z-eigenvalues is sometimes
not its largest US-eigenvalue, see Table 2.
Question 1: What is the necessary and sufficient condition for
the equality of the largest absolute Z-eigenvalue and
the largest US-eigenvalue to a real symmetric tensor?
谢谢大家！