Iterative Methods for Computing the Moore-Penrose Pseudoinverse of Quaternion Matrices, with Applications

2025/08/23 by Leplat, Valentin, Ahmadi-Asl, Salman, Pan, JunJun +1
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2508.16979

Abstract

We develop quaternion--native iterative methods for computing the Moore--Penrose (MP) pseudoinverse of quaternion matrices and analyze their convergence. Our starting point is a damped Newton--Schulz (NS) iteration tailored to noncommutativity: we enforce the appropriate left/right identities for rectangular inputs and prove convergence directly in ℍ under a simple spectral scaling. We then derive higher--order (hyperpower) NS schemes with exact residual recurrences that yield order-p local convergence, together with factorizations that reduce the number of s× s quaternion products per iteration. Beyond NS, we introduce a randomized sketch--and--project method (RSP--Q), a hybrid RSP+NS scheme that interleaves inexpensive randomized projections with an exact hyperpower step, and a matrix--form conjugate gradient on the normal equations (CGNE--Q). All algorithms operate directly in ℍ (no real or complex embeddings) and are matrix--free.

Citations

• Pass-efficient Randomized Algorithms for Low-rank Approximation of Quaternion Matrices
• Randomized Large-Scale Quaternion Matrix Approximation: Practical Rangefinders and One-Pass Algorithm
• Efficient quaternion CUR method for low-rank approximation to quaternion matrix
• Quaternion Knowledge Graph Embeddings
• Quaternion Convolutional Neural Networks for Heterogeneous Image Processing
• On the perturbation of the Moore-Penrose inverse of a matrix
• Randomized Quasi-Newton Updates are Linearly Convergent Matrix Inversion Algorithms
• Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
• Quaternions and matrices of quaternions
• Preliminary Sketch of Biquaternions

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• Pass-efficient Randomized Algorithms for Low-rank Approximation of Quaternion Matrices
• Generalizing Frobenius Inversion to Quaternion Matrices
• Quaternion matrix decomposition and its theoretical implications