A randomized singular value decomposition for third-order oriented tensors

TL;DR

This paper introduces a more efficient randomized singular value decomposition method for third-order oriented tensors, combining tensor-train decomposition with probabilistic algorithms to improve speed and maintain accuracy.

Contribution

It develops a novel randomized O-SVD algorithm for third-order tensors, enhancing efficiency and providing detailed error analysis.

Findings

The randomized O-SVD achieves comparable accuracy to existing methods.

Numerical experiments demonstrate improved efficiency in real-world applications.

The method effectively balances speed and precision in tensor decompositions.

Abstract

The oriented singular value decomposition (O-SVD) proposed by Zeng and Ng provides a hybrid approach to the t-product based third-order tensor singular value decomposition with the transform matrix being a factor matrix of the higher order singular value decomposition. Continuing along this vein, this paper explores realizing the O-SVD more efficiently by the tensor-train rank-1 decomposition and gives a truncated O-SVD. Motivated by the success of probabilistic algorithms, we develop a randomized version of the O-SVD and present its detailed error analysis. The new algorithm has advantages in efficiency while keeping good accuracy compared with the current tensor decompositions. Our claims are supported by numerical experiments on several oriented tensors from real applications.