Randomized Quaternion Singular Value Decomposition for Low-Rank Approximation

TL;DR

This paper introduces a randomized quaternion SVD algorithm for low-rank approximation, leveraging quaternion normal distribution for efficient data projection, with proven accuracy and improved performance in color face recognition tasks.

Contribution

The paper develops a novel randomized QSVD method using quaternion normal distribution, providing theoretical error bounds and demonstrating superior recognition accuracy and efficiency.

Findings

Achieves accurate low-rank approximation of quaternion matrices.

Outperforms existing Lanczos-based QSVD and fast directions algorithms.

Enhances color face recognition accuracy and computational efficiency.

Abstract

This paper presents a randomized quaternion singular value decomposition (QSVD) algorithm for low-rank matrix approximation problems, which are widely used in color face recognition, video compression, and signal processing problems. With quaternion normal distribution based random sampling, the randomized QSVD algorithm projects a high-dimensional data to a low-dimensional subspace and then identifies an approximate range subspace of the quaternion matrix. The key statistical properties of quaternion Wishart distribution are proposed and used to perform the approximation error analysis of the algorithm. Theoretical results show that the randomized QSVD algorithm can trace dominant singular value decomposition triplets of a quaternion matrix with acceptable accuracy. Numerical experiments also indicate the rationality of proposed theories. Applied to color face recognition problems, the randomized QSVD algorithm obtains higher recognition accuracies and behaves more efficient than the known Lanczos-based partial QSVD and a quaternion version of fast frequent directions algorithm.