Efficient Low-Rank Approximation of Quaternion Matrices via Adaptive Cross Approximation

Abstract

The adaptive cross approximation (ACA) algorithm is a powerful tool for constructing low-rank matrix approximations using only a small subset of matrix entries. While extensively studied for real and complex matrices, its extension to quaternion algebras, remains largely unexplored. This paper presents the first comprehensive framework for adaptive cross approximation operating natively in the quaternion algebra H. Unlike approaches that rely on real embeddings or complexifications, our method preserves the algebraic structure of quaternions throughout the computation, avoiding information loss and maintaining the natural correlations between quaternion components. We develop a fully quaternionic ACA algorithm with rigorous theoretical foundations, including convergence analysis, error bounds, and complexity estimates. The algorithm achieves O(r(m+n)) complexity per iteration and demonstrates quasioptimal approximation properties comparable to the quaternion singular value decomposition (QSVD), but with significantly reduced computational cost. We provide a detailed MATLAB with numerical experiments on color image compression. Our results show that native quaternion ACA outperforms significantly the basic SVD algorithm in terms of computational efficiency, with very close approximation accuracy.