A Randomized Tensor Singular Value Decomposition based on the t-product

TL;DR

This paper introduces a randomized tensor SVD method based on the t-product, offering a more computationally efficient approach for large third-order tensors with promising applications in data compression and analysis.

Contribution

It extends randomized matrix SVD techniques to third-order tensors, providing a faster, scalable factorization method with theoretical guarantees.

Findings

The proposed method achieves comparable accuracy to traditional t-SVD.

Numerical experiments demonstrate efficiency on large datasets.

Theoretical analysis confirms the method's reliability.

Abstract

The tensor Singular Value Decomposition (t-SVD) for third order tensors that was proposed by Kilmer and Martin~\cite{2011kilmer} has been applied successfully in many fields, such as computed tomography, facial recognition, and video completion. In this paper, we propose a method that extends a well-known randomized matrix method to the t-SVD. This method can produce a factorization with similar properties to the t-SVD, but is more computationally efficient on very large datasets. We present details of the algorithm, theoretical results, and provide numerical results that show the promise of our approach for compressing and analyzing datasets. We also present an improved analysis of the randomized subspace iteration for matrices, which may be of independent interest to the scientific community.