Biswarup Karmakar

Biswarup Karmakar, Neha Bhadala, Ratikanta Behera

The computation of generalized inverses of quaternion matrices is a fundamental problem in quaternion linear algebra, with wide-ranging applications in signal processing, image restoration, and multidimensional data analysis. This paper presents three efficient quaternion iterative algorithms for computing the Moore-Penrose pseudoinverse: (i) the quaternion rapid iterative method (QRAPID), (ii) the quaternion strong approximate inverse (QSAI), and (iii) the quaternion hyperpower iterative method of order nineteen (QHPI19). Convergence theorems and perturbation bounds are established to ensure numerical stability and robustness. The QSAI method is further employed as a preconditioner for quaternion Krylov subspace solvers, resulting in substantial reductions in the iteration count and runtime for large-scale linear systems. Comprehensive numerical experiments demonstrate that the proposed algorithms achieve an accuracy comparable to or better than existing approaches, including quaternion SVD, quaternion Newton-Schulz, and classical hyperpower schemes, while offering significant computational savings. The practical utility of the framework is illustrated through two representative applications: image completion via CUR decomposition and signal filtering, which confirm its scalability and effectiveness in real-world multidimensional data applications.

Biswarup Karmakar, Ratikanta Behera

Tensor completion has emerged as a powerful framework for recovering missing data in multidimensional signals by exploiting low-rank tensor structures. Among existing approaches, linear transform-based tensor nuclear norm (TNN) methods have achieved considerable success by enforcing low-rankness on transformed frontal slices. However, the low-rank structure revealed by linear transforms remains inherently limited. To better capture intrinsic correlations, nonlinear transform-based TNN (NTTNN) models have been proposed, significantly enhancing low-rank representation through composite transforms. Despite their effectiveness, existing NTTNN methods are restricted to real-valued tensors and fail to model quaternion-valued data, which are essential for preserving inter-channel dependencies in color images and videos. Extending nonlinear TNN models to the quaternion domain is challenging due to the non-commutativity of quaternion multiplication and the complexity of quaternion singular value decomposition. To address the limitations encountered in prior works, we propose a quaternion nonlinear transform-induced tensor nuclear norm (QNTTNN) via a real embedding of quaternions, enabling tractable nuclear norm definitions and efficient optimization. Building upon QNTTNN, we formulate a quaternion tensor completion model and develop a proximal alternating minimization algorithm with rigorous convergence guarantees. Extensive experiments on benchmark color video inpainting datasets validate the superior performance of the proposed method over existing approaches.

Biswarup Karmakar, Ratikanta Behera

The robust low-rank tensor completion problem addresses the challenge of recovering corrupted high-dimensional tensor data with missing entries, outliers, and sparse noise commonly found in real-world applications. Existing methodologies have encountered fundamental limitations due to their reliance on uniform regularization schemes, particularly the tensor nuclear norm and $\ell_1$ norm regularization approaches, which indiscriminately apply equal shrinkage to all singular values and sparse components, thereby compromising the preservation of critical tensor structures. The proposed tensor weighted correlated total variation (TWCTV) regularizer addresses these shortcomings through an $M$-product framework that combines a weighted Schatten-$p$ norm on gradient tensors for low-rankness with smoothness enforcement and weighted sparse components for noise suppression. The proposed weighting scheme adaptively reduces the thresholding level to preserve both dominant singular values and sparse components, thus improving the reconstruction of critical structural elements and nuanced details in the recovered signal. Through a systematic algorithmic approach, we introduce an enhanced alternating direction method of multipliers (ADMM) that offers both computational efficiency and theoretical substantiation, with convergence properties comprehensively analyzed within the $M$-product framework.Comprehensive numerical evaluations across image completion, denoising, and background subtraction tasks validate the superior performance of this approach relative to established benchmark methods.