A Randomized Block Krylov Method for Tensor Train Approximation

TL;DR

This paper introduces a randomized block Krylov method for tensor train approximation that effectively handles noisy, large-scale tensor data, with theoretical guarantees and demonstrated superior performance.

Contribution

It proposes a novel randomized algorithm based on block Krylov subspace iteration for tensor train approximation, suitable for noisy data, with proven theoretical guarantees.

Findings

Effective handling of heavy-tailed noisy data

Theoretical guarantees for the proposed algorithm

Successful numerical experiments on synthetic and real data

Abstract

Tensor train decomposition is a powerful tool for dealing with high-dimensional, large-scale tensor data, which is not suffering from the curse of dimensionality. To accelerate the calculation of the auxiliary unfolding matrix, some randomized algorithms have been proposed; however, they are not suitable for noisy data. The randomized block Krylov method is capable of dealing with heavy-tailed noisy data in the low-rank approximation of matrices. In this paper, we present a randomized algorithm for low-rank tensor train approximation of large-scale tensors based on randomized block Krylov subspace iteration and provide theoretical guarantees. Numerical experiments on synthetic and real-world tensor data demonstrate the effectiveness of the proposed algorithm.