# Invariant Theory by M. Neusel

By M. Neusel

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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.