Algebraic Operations on Tensor Trains

2026-06-12

AI Summary

This work addresses the curse of dimensionality that severely hampers high-dimensional tensor computations, where conventional methods struggle to balance computational efficiency and memory consumption. Building upon the Tensor Train (TT) or Matrix Product State (MPS) representation, the paper introduces specialized algebraic algorithms for efficiently performing vector and matrix addition, Hadamard products, and MPO–MPS multiplication. The proposed approach achieves a superior trade-off among computational complexity, memory footprint, and numerical accuracy compared to existing techniques, thereby substantially enhancing the practicality of high-dimensional tensor operations. This advancement is particularly beneficial for memory-constrained applications such as system identification and dynamic programming.

Abstract

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.

Problem

• tensor train
• algebraic operations
• high-dimensional tensors
• matrix-vector product
• Hadamard product

Innovation

• Tensor Train
• MPO-MPS
• Hadamard product
• matrix-vector product
• computational efficiency