Algebraic Operations on Tensor Trains
For researchers and practitioners using tensor train decompositions in high-dimensional data analysis, the proposed algorithms offer more efficient operations without sacrificing accuracy.
This paper introduces algorithms for summation, Hadamard product, and matrix-vector product of tensors in the tensor train format, achieving an improved tradeoff between computational efficiency and accuracy compared to state-of-the-art methods.
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.