Quaternion deep matrix factorization and non-local Laplacian regularization for matrix completion

Quaternion-based methods for encoding multi-dimensional data have garnered increasing attention in recent years. However, explicitly constructing low-rank regularization terms for quaternion matrices typically relies on quaternion singular value decomposition (QSVD), which is computationally intensive. To address this limitation, we propose the quaternion deep matrix factorization (QDMF) model, which implicitly embeds the quaternion matrix rank information into the fidelity term without relying on QSVD. Furthermore, we introduce the quaternion non-local Laplacian and propose two regularization terms: the quaternion Dirichlet energy (QDE) and quaternion non-local curvature (QNC). These terms promote low-rankness of quaternion matrices by explicitly smoothing the matrices. Experimental results on both simulated and real data demonstrate that the proposed models outperform state-of-the-art quaternion models.