Conference abstracts

Session B7 - Numerical Linear Algebra

July 13, 15:30 ~ 16:00 - Room B5

Orthogonal tensors and their extremal spectral properties

André Uschmajew

University of Bonn, Germany   -

Let $\| X \|_2$, $\| X \|_F$ and $\| X \|_*$ denote the spectral, Frobenius and nuclear norm of an $n_1 \times \dots \times n_d$ tensor $X$. The sharp upper bounds for $\| X \|_F / \| X \|_2$ and $\| X \|_* / \| X \|_F$ seem to be unknown for many configurations of dimensions $(n_1,\dots,n_d)$ with $d \ge 3$. Assuming $n_1 \le \dots \le n_d$, a trivial upper bound for both ratios is $\sqrt{n_1 \cdots n_d}$. It is sharp for matrices ($d=2$) and achieved if and only if $X$ is a multiple of a matrix with pairwise orthonormal rows. Using a natural definition of orthogonal higher-order tensors, we can generalize this fact in the sense that orthogonality is necessary (up to scaling) and sufficient to achieve the trivial upper bound for both ratios. However, as it turns out, orthogonal tensors do not necessarily exist for every configuration $(n_1,\dots,n_d)$. When $d=3$, the question of existence of real orthogonal tensors admits an equivalent algebraic formulation known as Hurwitz' problem on composition formulas for bilinear forms. A classical result (due to Hurwitz) then implies that real $n \times n \times n$ orthogonal tensors only exist for $n = 1,2,4,8$. Surprisingly, the situation is different in complex spaces: unitary $n \times n \times n$ do not exist for any $n \ge 2$.

Joint work with Zhening Li (University of Portsmouth), Yuji Nakatsukasa (Oxford University) and Tasuku Soma (University of Tokyo).

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FoCM 2017, based on a nodethirtythree design.