A Parallelizable Quaternion Higher-Order Singular Value Decomposition with Applications

TL;DR

This paper introduces a parallelizable quaternion higher-order singular value decomposition (TS-QHOSVD) that maintains key properties of HOSVD, enabling efficient tensor data analysis and applications like video denoising and scientific data compression.

Contribution

It proposes a novel two-sided quaternion HOSVD that can be parallelized, preserving important properties and providing error bounds, with demonstrated effectiveness in practical applications.

Findings

TS-QHOSVD preserves HOSVD ordering and orthogonality properties.

The truncated TS-QHOSVD has a proven error bound.

Applications show improved efficiency in video denoising and scientific data compression.

Abstract

Higher-order singular value decomposition (HOSVD) is a celebrated tool for tensor data analysis. The sequential HOSVD was recently generalized to the quaternion domain, while a naive quaternion extension of the classical HOSVD% by De Lathauwer et al., which can be excecuted in parallel, incurs issues. To leverage the power of parallel computing, this work introduces a two-sided quaternion HOSVD (TS-QHOSVD) that can be parallelized on two processors. It is proved that TS-QHOSVD (i) preserves the HOSVD ordering property, (ii) inherits the orthogonality property at the first and the last modes, and (iii) satisfies the weak orthogonality at all modes. The truncated TS-QHOSVD is then developed, with its error bound being established. We apply the proposed model on color video denoising as well as scientific data compression arising from 3D Navier-Stokes equation and Lorentz system to demonstrate its efficacy.