Iterative Methods for Computing the Moore-Penrose Pseudoinverse of Quaternion Matrices, with Applications
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915540593803264 |
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| author | Leplat, Valentin Ahmadi-Asl, Salman Pan, JunJun Zheng, Ning |
| author_facet | Leplat, Valentin Ahmadi-Asl, Salman Pan, JunJun Zheng, Ning |
| contents | 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 $\mathbb{H}$ under a simple spectral scaling. We then derive higher--order (\emph{hyperpower}) NS schemes with exact residual recurrences that yield order-$p$ local convergence, together with factorizations that reduce the number of $s\times s$ 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 $\mathbb{H}$ (no real or complex embeddings) and are matrix--free. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iterative Methods for Computing the Moore-Penrose Pseudoinverse of Quaternion Matrices, with Applications Leplat, Valentin Ahmadi-Asl, Salman Pan, JunJun Zheng, Ning Numerical Analysis 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 $\mathbb{H}$ under a simple spectral scaling. We then derive higher--order (\emph{hyperpower}) NS schemes with exact residual recurrences that yield order-$p$ local convergence, together with factorizations that reduce the number of $s\times s$ 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 $\mathbb{H}$ (no real or complex embeddings) and are matrix--free. |
| title | Iterative Methods for Computing the Moore-Penrose Pseudoinverse of Quaternion Matrices, with Applications |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.16979 |