A Randomized Tensor Train Singular Value Decomposition

TL;DR

This paper introduces a randomized algorithm for computing the hierarchical SVD in the tensor train format, enabling efficient low-rank tensor approximations in high-dimensional data analysis.

Contribution

It extends randomized matrix decomposition techniques to tensor train structures, providing a novel and efficient method for hierarchical tensor SVD computation.

Findings

The algorithm achieves quasi-best low-rank approximations.

It reduces computational costs compared to traditional methods.

The approach is applicable to high-dimensional tensor data.

Abstract

The hierarchical SVD provides a quasi-best low rank approximation of high dimensional data in the hierarchical Tucker framework. Similar to the SVD for matrices, it provides a fundamental but expensive tool for tensor computations. In the present work we examine generalizations of randomized matrix decomposition methods to higher order tensors in the framework of the hierarchical tensors representation. In particular we present and analyze a randomized algorithm for the calculation of the hierarchical SVD (HSVD) for the tensor train (TT) format.