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1、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,张量的基本概念 张量特征值的计算 张量秩1逼近和低秩逼近 张量计算软件 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,1阶张量：向量,2阶张量：矩阵 A=(aij),3阶张量：长方体 A=(aijk),张量的秩,张量的秩： 1927年 Hitchcock,NP-Hard,n-rank,秩1张量：,可计算,其中 表示 张量X的mode-k mode,秩1矩阵：A=a bT = (aibj),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,Tucker逼近,
2、最佳秩1逼近：R=1,1. 张量的基本概念,1. 张量的基本概念,张量的完备化,低秩张量M部分元素 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,Z（E）-特征值,H-特征值,US-特征值,2005, Qi,B-特征值,2014，Cui, Dai, Nie,2014，Ni, Qi, Bai,张量的特征值,1. 张量的基本概念,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：,Ng, Qi, Zhou 2009， Chang, Pearson, Zhang 2011， L. Zhang, L. Qi 2012， Qi, Q. Ya
3、ng, Y. Yang 2013,Perron-Frobenius 理论,对称张量的最大Z-特征值的计算：,The sequential SDPs method Hu, Huang, Qi 2013 Sequential subspace projection methodHao, Cui, Dai. 2014 Shifted symmetric higher-order power method Kolda,Mayo 2011 Jacobian semidefinite relaxations 计算对称张量所有实的B-特征值 Cui, Dai, Nie 2014,对称张量的US-特征值的计算
4、：,Geometric measure of entanglement and U-eigenvalues of tensors, SIAM Journal on Matrix Analysis and Applications，Ni，Qi，Bai 2014 Complex Shifted Symmetric higher-order power method Ni, Bai 2014,2. 张量特征值的计算,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
5、，或稳定点,Nie, Wang2013 ：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，Bai2014 ：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, Te
6、nsor 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,A Guyan Ni, Liqun Qi and Minru Bai, Geometric measure of entanglement and U-eigenvalues of tensors, SIAM
7、 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.,5. 复张量的最佳秩1逼近和特征值,Basic Definitions,1. A tensor S is called symmetric as its entries s_i1id are invariant under any permu
8、tation of their indices.,2. A Z-eigenpair (, u) to a real symmetric tensor S is defined by,3. An eigenpair (, u) to a real symmetric tensor S is defined by,2005, Qi,2011, Kolda and Mayo,7 T.G. Kolda and J.R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis
9、and Applications, 32(2011), pp. 1095-1124.,uTu,u*Tu,4. The best rank-one tensor approximation problems,Assume that T a d-order real tensor. Denote a rank-one tensor,is to minimizes the least-squares cost function,. Then the rank-one approximation problem,The rank-one tensor,rank-one approximation to
10、 tensor T.,is said to be the best real,If T is a symmetric real tensor,the best real symmetric rank-one approximation.,is said to be,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,
11、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 symmetr
12、ic, 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
13、 d-order complex tensor will be denoted by,inner product,norm,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,The superscript * denotes the complex conjugate. The superscript T denotes transposi
14、tion.,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 (US-e
15、igenvalue) of S if and a nonzero vector,The largest | 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
16、 d-order complex tensor S can also be defined by the following equation system,or,(1),3.1. US-eigenpairs of symmetric tensors,Theorem 3. (Takagis factorization) Let A Cnn be a complex symmetric tensor. Then there exists a unitary matrix U Cnn such that,Case d=2:,Theorem 4. Let A Cnn be a complex sym
17、metric tensor. Let U Cnn be a unitary matrix such that,Let ei = (0, , 0, 1, 0, , 0)T , i = 1, , n.,Then both,and,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-e
18、igenvectors with respect to 1 is,3.2. US-eigenpairs of symmetric tensors,The problem of finding eigenpairs is equivalent to solving a polynomial system,Case d 3,8 S. Friedland, Best rank one approximation of real symmetric tensors can be chosen symmetric, Frontiers of Mathematics in China, 8(2013),
19、pp. 19-40.,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,
20、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).,Case d 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
21、 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).,3.2. US-eigenpairs of symmetric
22、 tensors,Note. 1. Let S be the symmetric 2 2 2 2 tensor whose non-zero 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
23、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 d1 = x has infinite non-zero solutions, where S is a s
24、ymmetric 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 333 tensor as in Note 2. Then
25、x = for all 0 a 1 are non-zero solutions of Sx d1 = 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 US-eigenvalue of S. Then a) is also a US-eigenvalue of S ; b) G(S)
26、= max.,Case d=2,Theorem 8. Let A Cnn 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,
27、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,approximation of S for each,rank-one,The problem of finding eigenpairs is equivale
28、nt to solving a polynomial system,Let x = y + z1, y, z Rn. Then Q3 is equivalent to the following problem,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
29、, 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.,The best real rank-one approximation is sometimes also the best complex rank-one approximation even if the tensor is not a s
30、ymmetric nonnegative real tensor, see Table 1. The absolute-value largest of Z-eigenvalues is sometimes not its largest US-eigenvalue, see Table 2.,By observing numerical examples, we find the following results:,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！,