The tensor train (TT) model is widely used to approximate high-dimensional tensors, enabling efficient handling of data that may exceed available memory. TT helps address the curse of dimensionality in applications such as system identification and dynamic programming. In some applications, TT is known as a ``matrix product state" (MPS). This paper introduces algorithms that facilitate the summation, Hadamard (elementwise) product, and matrix--vector product of matrices and vectors (tensors) represented in the tensor train (TT) format. The last product is also known under the acronym MPO--MPS. The proposed algorithms achieve an improved tradeoff between computational efficiency and accuracy compared to state-of-the-art methods.

翻译：张量列（TT）模型广泛应用于高维张量的近似，使得能够高效处理可能超出内存容量的数据。在系统辨识和动态规划等应用中，TT有助于解决维度灾难问题。在某些应用场景中，TT也被称为“矩阵乘积态”（MPS）。本文提出了若干算法，用于实现以张量列（TT）格式表示的矩阵与向量（张量）之间的求和运算、哈达玛（逐元素）乘积以及矩阵-向量乘积。其中最后一种乘积也以缩写MPO-MPS著称。与现有最优方法相比，所提出的算法在计算效率与精度之间实现了更优的权衡。