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

TL;DR

The paper addresses computing the Moore–Penrose pseudoinverse for quaternion matrices by developing quaternion-native iterative schemes that respect noncommutativity. It introduces damped and higher-order Newton–Schulz updates, cost-saving factorizations, and two complementary approaches: a randomized sketch–and–project method and a matrix–form CGNE variant, all working directly in $D$ without embeddings. Theoretical results establish convergence in the quaternion setting for full column/row ranks, with exact residual recurrences and local order-$p$ convergence for hyperpower NS. Empirical benchmarks across random matrices and applications (CUR-based completion, Lorenz filtering, quaternion FFT-based deblurring) show that NS variants deliver strong accuracy and speed, while RSP–Q and CGNE–Q offer competitive alternatives; the methods enable scalable, structure-preserving quaternion linear algebra for large-scale problems.

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 $\mathbb{H}$ 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.

Theorems & Definitions (2)

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• proof