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

TL;DR

This paper introduces quaternion-native iterative algorithms, including Newton-Schulz variants and hybrid methods, for efficiently computing the Moore-Penrose pseudoinverse of quaternion matrices directly in quaternion space.

Contribution

It develops and analyzes quaternion-native iterative methods, including higher-order schemes and hybrid algorithms, for computing the Moore-Penrose pseudoinverse without embedding into real or complex spaces.

Findings

Convergence proven directly in quaternion space.

Higher-order methods achieve faster local convergence.

Hybrid randomized and deterministic schemes improve efficiency.

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.