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 new, direct estimate of the entrywise approximation error that is applicable in such cases. Our argument is based on random embeddings, the tensor-structured Hanson--Wright inequality, and the core coherences of a tensor. The theoretical results are accompanied with numerical experiments: we compute low-rank tensor-train approximations in the maximum norm for two classes of tensors using the method of alternating projections.

翻译：当近似误差以Frobenius范数度量时，低秩张量列车近似的理论已得到充分理解。逐元素最大范数同样重要，但对于大规模张量而言其约束力显著减弱，导致通过Frobenius范数及范数等价性获得的估计结果过于悲观甚至失去意义。本文针对此类情形推导了一种新的逐元素近似误差直接估计方法。该论证基于随机嵌入、张量结构化的Hanson-Wright不等式以及张量的核相干性。理论结果配有数值实验验证：我们采用交替投影方法，在最大范数意义下对两类张量分别计算低秩张量列车近似。