Quaternion adaptive approximation normalization graph guided implicit low rank for robust matrix completion

Graph structures are effective for capturing low-dimensional manifolds within high-dimensional data spaces and are frequently utilized as regularization terms to smooth graph signals. A crucial element in this process is the construction of the graph Laplacian. However, the normalization of this Laplacian often necessitates computationally expensive inverse operations. To address this limitation, this paper introduces quaternion graph regularity and proposes the quaternion adaptive approximation normalization graph (QAANG). QAANG offers a computationally efficient solution by requiring only a single adaptive scalar for approximate normalization, thereby circumventing the need for inverse operations. To promote the low rank of the graph, we implicitly embed the low rank into the data fidelity term. This approach not only avoids the significant costs associated with the explicit computation of the low-rank of quaternion matrices, but also eliminates the need to balance multiple regularization terms and adjust hyperparameters. Experimental results demonstrate that QAANG surpasses current state-of-the-art quaternion methods in both completion performance and robustness.