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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,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
2、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. Yang, Y. Yang 2013,Perron-Frobenius 理论,对称张量的最大Z-特征值的计算：,The seque
3、ntial 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：,Geometric measure of entanglement and U-eigenvalues of tensor
4、s, 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，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，Bai201
5、4 ：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。
6、 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 Journal on Matrix Analysis and Applications 2014, 35(1): 73-87
7、,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 permutation of their indices.,2. A Z-eigenpair (, u) to a real symme
8、tric 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 and Applications, 32(2011), pp. 1095-1124.,uTu,u*Tu,4. The best
9、 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 tensor T.,is said to be the best real,If T is a symmetric real
10、 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, can be chosen symmetric., ud is the best real rank-one approxim
11、ation 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.
12、 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,1
13、0 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 transposition.,For A,B H, define the inner product and norm as,inner pro
14、duct,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-eigenvalue) of S if and a nonzero vector,The largest | is the en
15、tanglement 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
16、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 symmetric 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,
