The theory of low-rank tensor-train approximation is well understood when the approximation error is measured in the Frobenius norm. The entrywise maximum norm is equally important but is significantly weaker for large tensors, making the estimates obtained via the Frobenius norm and norm equivalence pessimistic or even meaningless. In this article, we derive a direct estimate of the entrywise approximation error that is applicable in some of these cases. The estimate is given in terms of the higher-order generalization of the matrix factorization norm, and its proof is based on the tensor-structured Hanson--Wright inequality. The theoretical results are accompanied by numerical experiments carried out with the method of alternating projections.

翻译：当逼近误差以Frobenius范数度量时，低秩张量链逼近理论已较为完善。逐项最大范数同样重要，但对于大规模张量而言其约束力显著弱于Frobenius范数，导致通过范数等价性从Frobenius范数导出的估计过于保守甚至失去意义。本文针对此类情形，推导出可直接应用于逐项逼近误差的估计方法。该估计通过矩阵分解范数的高阶推广形式给出，其证明基于张量结构的Hanson-Wright不等式。理论结果辅以交替投影法进行的数值实验验证。