Theory and Computation of Tensors. Multi-Dimensional Arrays by Yimin Wei, Weiyang Ding

By Yimin Wei, Weiyang Ding

Theory and Computation of Tensors: Multi-Dimensional Arrays investigates theories and computations of tensors to develop views on matrices. facts within the monstrous information period isn't just turning out to be greater but additionally turning into even more complex. Tensors (multi-dimensional arrays) come up obviously from many engineering or medical disciplines simply because they could symbolize multi-relational info or nonlinear relationships.

  • Provides an advent of contemporary effects approximately tensors
  • Investigates theories and computations of tensors to expand views on matrices
  • Discusses easy methods to expand numerical linear algebra to numerical multi-linear algebra
  • Offers examples of the way researchers and scholars can have interaction in study and the functions of tensors and multi-dimensional arrays

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H2 (b) H:,:,2 =   . .  .. 2) .. .. HJ1 HJ1 +1 · · · HJ1 +J2 −2 .. .. .. . .  HJ3 −1 HJ3 ··· HJ2 +J3 −3  . . .  . .. HJ3 (b) H:,:,J3 =   .. .  .. .. HJ1 +J3 −2 HJ1 +J3 −1 · · · HJ1 +J2 +J3 −4  HJ2 +J3 −2  ..  . .  HJ1 +J2 +J3 −4  HJ1 +J2 +J3 −3 Then the BHHB tensor H has the level-2 Vandermonde decomposition ⊗ Z1,1 ) ×2 (Z2,2 ⊗ Z1,2 ) · · · ×m (Z2,m ⊗ Z1,m ) , H = C ×1 (Z2,1 KR KR KR 44 CHAPTER 3. FAST TENSOR-VECTOR PRODUCTS where C is a diagonal tensor with diagonal entries {ck }K k=1 , each a Vandermonde matrix    Ip −1 2 2 1 z1,1 z1,1 · · · z1,1 1 z2,1 z2,1   Ip −1  2 2 1 z1,2 z1,2 · · · z1,2  1 z2,2 z2,2  Z1,p =  ..

Let A be a complex tensor, E be a nonnegative tensor, b be a complex vector, and d be a nonnegative vector. Define Σ := x ∈ Cn : A + E xm−1 = b + d, E ≤ E, d ≤ d . Then Σ = x ∈ Cn : Axm−1 − b ≤ E|x|m−1 + d . Proof. On the one hand, if x ∈ Σ, then Axm−1 − b = −Exm−1 + d ≤ E|x|m−1 + d. On the other hand, if Axm−1 − b ≤ E|x|m−1 + d, then there exist two signature matrices S1 and S2 such that S1 Axm−1 − b = Axm−1 − b and S2 x = |x|, which indicates that S1 Axm−1 − b ≤ ES2m−1 xm−1 + d. Thus there is a diagonal matrix D with |D| ≤ I such that Axm−1 − b = S1∗ DES2m−1 xm−1 + S1∗ Dd.

We first construct • 3rd -order square Hankel tensors of size n × n × n (n = 10, 20, . . , 100), and • 3rd -order square BHHB tensors of level-1 size n1 × n1 × n1 and level-2 size n2 × n2 × n2 (n1 , n2 = 5, 6, . . , 12). Then we compute the tensor-vector products H ×2 x2 ×3 x3 using • our proposed fast algorithm based on FFT, and • the non-structured algorithm based on the definition. The average running times of 1000 products are shown in Fig. 2. From the results, we can see that the running time of our algorithm increases far more slowly than that of the non-structured algorithm just as predicted by the theoretical analysis.

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