Many problems in modern scientific computing are challenging because of a \emph{curse of dimension}, where their mathematical formulation involves objects whose dimension is \emph{exponential} in the nominal "size" of the problem. Tensor networks can provide a compact representation for exponentially large vectors and matrices that arise in applications, but these representations do not always lead to reliable algorithms. This paper develops and analyzes techniques for randomized dimension reduction of tensor network data. These techniques support a suite of efficient algorithms for provably solving exponential-scale linear algebra problems, including trace estimation and eigenvalue approximation. The paper includes several stylized illustrations from quantum many-body physics with ambient dimension up to $2^{200}$.

翻译：现代科学计算中的许多问题因“维度灾难”而颇具挑战性，其数学表述涉及的对象维度在名义“规模”上呈指数级增长。张量网络可为应用中出现的指数级庞大向量和矩阵提供紧凑表示，但这类表示未必总能导出可靠的算法。本文开发并分析了张量网络数据的随机降维技术。这些技术支撑起一套高效算法套件，可用于可证明地解决指数级规模的线性代数问题，包括迹估计和特征值逼近。文中还通过环境维度高达$2^{200}$的量子多体物理中的若干示例进行了阐释。