标题： 树张量网络格式中流式传输的随机低秩逼近算法 显示英文标题

标题： Randomized algorithms for streaming low-rank approximation in tree tensor network format

摘要： 在本工作中，我们提出了树张量网络Nyström（TTNN）算法，该算法扩展了最近关于可流式张量近似的研究，例如针对Tucker和张量-列车格式的研究，以更一般的树张量网络格式，从而实现了对各种现有方法的统一处理。 我们的方法保留了矩阵广义Nyström近似的关键特征，即随机化、单次遍历、可流式和成本效益高。 此外，采样结构允许并行实现。 我们为该算法提供了确定性误差界，在特定的高斯维度约简映射情况下，还提供了概率性误差界。 我们还引入了该算法的顺序变体，称为顺序树张量网络Nyström（STTNN），该变体在密集张量上表现更好。 此外，这两种算法都适用于树张量网络格式中张量的重新压缩或舍入。 数值实验突显了所提出方法的效率和有效性。 显示英文摘要

摘要： In this work, we present the tree tensor network Nystr\"om (TTNN), an algorithm that extends recent research on streamable tensor approximation, such as for Tucker and tensor-train formats, to the more general tree tensor network format, enabling a unified treatment of various existing methods. Our method retains the key features of the generalized Nystr\"om approximation for matrices, that is randomized, single-pass, streamable, and cost-effective. Additionally, the structure of the sketching allows for parallel implementation. We provide a deterministic error bound for the algorithm and, in the specific case of Gaussian dimension reduction maps, also a probabilistic one. We also introduce a sequential variant of the algorithm, referred to as sequential tree tensor network Nystr\"om (STTNN), which offers better performance for dense tensors. Furthermore, both algorithms are well-suited for the recompression or rounding of tensors in the tree tensor network format. Numerical experiments highlight the efficiency and effectiveness of the proposed methods.