Randomized algorithms for computing the tensor train approximation and their applications

TL;DR

This paper introduces randomized algorithms for tensor train approximation that are faster than traditional methods, providing reliable rank estimation and accuracy comparable to existing algorithms, with applications demonstrated on synthetic and real data.

Contribution

The paper develops new randomized algorithms for fixed TT-rank tensor approximation, combining random projections with the power scheme and deriving theoretical bounds.

Findings

Algorithms are several times faster than TT-SVD.

Achieve comparable accuracy to TT-SVD.

Effective rank estimation strategies demonstrated.

Abstract

In this paper, we focus on the fixed TT-rank and precision problems of finding an approximation of the tensor train (TT) decomposition of a tensor. Note that the TT-SVD and TT-cross are two well-known algorithms for these two problems. Firstly, by combining the random projection technique with the power scheme, we obtain two types of randomized algorithms for the fixed TT-rank problem. Secondly, by using the non-asymptotic theory of sub-random Gaussian matrices, we derive the upper bounds of the proposed randomized algorithms. Thirdly, we deduce a new deterministic strategy to estimate the desired TT-rank with a given tolerance and another adaptive randomized algorithm that finds a low TT-rank representation satisfying a given tolerance, and is beneficial when the target TT-rank is not known in advance. We finally illustrate the accuracy of the proposed algorithms via some test tensors from synthetic and real databases. In particular, for the fixed TT-rank problem, the proposed algorithms can be several times faster than the TT-SVD, and the accuracy of the proposed algorithms and the TT-SVD are comparable for several test tensors.