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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s\times s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>×</mo> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation> (no real or complex embeddings) and avoid full decompositions of A. Numerically, we test the performance of the proposed algorithms on controlled synthetic matrices. Across these tests, the damped NS method provides the strongest overall accuracy/runtime trade-off among the iterative schemes considered. In three application case studies (CUR image/video completion, Lorenz filtering, FFT-based deblurring), we deploy only the NS family and obtain competitive accuracy and wall time while operating directly in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation>. These quaternion-native methods are suitable as drop-in solvers for large-scale quaternion inverse problems.

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

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发表日期：

2026年4月30日

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Valentin Leplat,
Salman Ahmadi-Asl,
JunJun Pan,
Ning Zheng

摘要

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 $\mathbb {H}$ $H$ 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 $s\times s$ $s \times s$ 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 $\mathbb {H}$ $H$ (no real or complex embeddings) and avoid full decompositions of A. Numerically, we test the performance of the proposed algorithms on controlled synthetic matrices. Across these tests, the damped NS method provides the strongest overall accuracy/runtime trade-off among the iterative schemes considered. In three application case studies (CUR image/video completion, Lorenz filtering, FFT-based deblurring), we deploy only the NS family and obtain competitive accuracy and wall time while operating directly in $\mathbb {H}$ $H$ . These quaternion-native methods are suitable as drop-in solvers for large-scale quaternion inverse problems.