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arxiv: 2605.01464 · v1 · submitted 2026-05-02 · 🧮 math.NA · cs.NA

A Family of Iterative Methods for Computing Generalized Inverses of Quaternion Matrices and its Applications

Biswarup Karmakar , Neha Bhadala , Ratikanta Behera This is my paper

Pith reviewed 2026-05-09 17:53 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords quaternion matricesMoore-Penrose pseudoinverseiterative methodsgeneralized inversespreconditioningKrylov solversimage completionsignal filtering
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The pith

Three new iterative algorithms compute the Moore-Penrose pseudoinverse of quaternion matrices with accuracy matching or exceeding existing methods at lower computational cost.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces three quaternion-specific iterative algorithms, QRAPID, QSAI, and QHPI19, to approximate the Moore-Penrose pseudoinverse of quaternion matrices. It establishes convergence results and perturbation bounds that guarantee stability under suitable conditions. Numerical tests show these algorithms perform at least as well as quaternion SVD, Newton-Schulz iterations, and classical hyperpower methods while requiring fewer operations, and the QSAI variant doubles as a preconditioner that cuts iteration counts in Krylov solvers. The work demonstrates the approach on image completion via CUR decomposition and on signal filtering tasks. If the claims hold, large-scale quaternion problems in multidimensional data become more tractable without sacrificing reliability.

Core claim

The paper claims that the QRAPID, QSAI, and QHPI19 iterative schemes converge to the Moore-Penrose pseudoinverse of any quaternion matrix for which the methods are defined, that their perturbation bounds ensure numerical robustness, and that extensive experiments confirm accuracy comparable to or better than quaternion SVD, Newton-Schulz, and hyperpower baselines together with substantial runtime reductions; the QSAI scheme additionally serves as an effective preconditioner for quaternion Krylov solvers, and the overall framework scales to practical tasks such as CUR-based image completion and signal filtering.

What carries the argument

The three quaternion iterative schemes (QRAPID, QSAI, and QHPI19) that generate successive matrix approximations to the Moore-Penrose pseudoinverse through repeated quaternion matrix multiplications and inversions of smaller blocks.

If this is right

  • The QSAI scheme reduces iteration counts and wall-clock time when used to precondition large quaternion Krylov systems.
  • CUR decomposition for image completion becomes feasible on larger quaternion-valued data sets without prohibitive cost.
  • Signal filtering pipelines that rely on generalized inverses run faster while preserving accuracy.
  • Perturbation bounds supply a priori guarantees on how small perturbations in the input matrix affect the computed inverse.
  • The methods serve as drop-in replacements for slower existing quaternion inversion routines in multidimensional applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same iterative pattern may extend to other generalized inverses such as the Drazin inverse or group inverse once the appropriate projector conditions are incorporated.
  • The preconditioning benefit observed with Krylov solvers could transfer to other iterative solvers that operate on quaternion blocks.
  • Runtime gains on current test matrices suggest that the approach could support real-time processing pipelines if the per-iteration cost continues to scale linearly with matrix dimension.

Load-bearing premise

The input quaternion matrices satisfy the algebraic conditions that guarantee convergence of the chosen iterative schemes and the test problems accurately represent the cost and accuracy behavior on large real-world instances.

What would settle it

A quaternion matrix of moderate size for which any of the three proposed methods diverges or produces an error larger than quaternion SVD, or a large-scale linear system where the QSAI-preconditioned Krylov solver requires more runtime than the unpreconditioned version or the direct SVD route.

read the original abstract

The computation of generalized inverses of quaternion matrices is a fundamental problem in quaternion linear algebra, with wide-ranging applications in signal processing, image restoration, and multidimensional data analysis. This paper presents three efficient quaternion iterative algorithms for computing the Moore-Penrose pseudoinverse: (i) the quaternion rapid iterative method (QRAPID), (ii) the quaternion strong approximate inverse (QSAI), and (iii) the quaternion hyperpower iterative method of order nineteen (QHPI19). Convergence theorems and perturbation bounds are established to ensure numerical stability and robustness. The QSAI method is further employed as a preconditioner for quaternion Krylov subspace solvers, resulting in substantial reductions in the iteration count and runtime for large-scale linear systems. Comprehensive numerical experiments demonstrate that the proposed algorithms achieve an accuracy comparable to or better than existing approaches, including quaternion SVD, quaternion Newton-Schulz, and classical hyperpower schemes, while offering significant computational savings. The practical utility of the framework is illustrated through two representative applications: image completion via CUR decomposition and signal filtering, which confirm its scalability and effectiveness in real-world multidimensional data applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes three iterative algorithms—QRAPID, QSAI, and QHPI19—for computing the Moore-Penrose pseudoinverse of quaternion matrices. It derives convergence theorems and perturbation bounds for these methods, employs QSAI as a preconditioner within quaternion Krylov subspace solvers, and reports numerical experiments claiming accuracy comparable or superior to quaternion SVD, Newton-Schulz iteration, and classical hyperpower schemes, together with runtime savings. Two applications (image completion via CUR decomposition and signal filtering) are used to illustrate practical utility.

Significance. If the convergence results and numerical claims hold, the work would supply practical, matrix-free alternatives to direct methods such as quaternion SVD for generalized-inverse computations. The preconditioning strategy and reported iteration reductions could be useful for large-scale quaternion linear systems arising in signal processing and multidimensional data analysis.

major comments (2)
  1. [§3] §3 (Convergence analysis): the statement that QRAPID converges for any initial matrix satisfying ||I - A X_0|| < 1 is not accompanied by an explicit, easily verifiable construction of such an X_0 for arbitrary quaternion matrices; without this, the practical scope of the claimed parameter-free convergence remains unclear.
  2. [§4.2] §4.2, Tables 1–3: the largest test matrices are 200 × 200; the reported iteration counts and runtime gains versus quaternion SVD therefore do not yet substantiate the scalability claim for the “large-scale linear systems” referenced in the abstract and in the Krylov-preconditioning application.
minor comments (2)
  1. The definition of the quaternion matrix norm used in the perturbation bounds should be stated explicitly at first use rather than only in the appendix.
  2. Figure 3 caption should indicate the matrix dimensions and conditioning numbers of the test problems shown.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will revise the manuscript to incorporate clarifications and additional experiments where needed.

read point-by-point responses

  1. Referee: [§3] §3 (Convergence analysis): the statement that QRAPID converges for any initial matrix satisfying ||I - A X_0|| < 1 is not accompanied by an explicit, easily verifiable construction of such an X_0 for arbitrary quaternion matrices; without this, the practical scope of the claimed parameter-free convergence remains unclear.

    Authors: We agree that an explicit construction for X_0 would improve the practical utility of the convergence result. In the revised manuscript, we will add a remark in §3 providing a concrete choice: X_0 = A^H / ||A||_F^2 (with A^H the quaternion conjugate transpose). We will include a short verification showing that this choice satisfies the norm condition for a wide class of matrices (or can be scaled by a small factor to ensure ||I - A X_0|| < 1), along with guidance on a posteriori checking of the condition. This addresses the concern without altering the theorem statement. revision: yes

  2. Referee: [§4.2] §4.2, Tables 1–3: the largest test matrices are 200 × 200; the reported iteration counts and runtime gains versus quaternion SVD therefore do not yet substantiate the scalability claim for the “large-scale linear systems” referenced in the abstract and in the Krylov-preconditioning application.

    Authors: The referee correctly notes that the current experiments use matrices up to 200 × 200. To strengthen the scalability claims, we will expand §4.2 with new experiments on larger matrices (up to at least 1000 × 1000) and report iteration counts and runtimes for the QSAI-preconditioned Krylov solvers on these sizes. The additional results will be added to the tables and discussed in the text to better support the large-scale claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; methods and theorems are independently derived

full rationale

The paper constructs QRAPID, QSAI, and QHPI19 as new iterative schemes for quaternion Moore-Penrose inverses, supported by explicitly stated convergence theorems and perturbation bounds derived from quaternion algebra properties. These derivations do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. Numerical comparisons and applications serve as external validation rather than inputs to the core claims. The derivation chain remains self-contained against the stated assumptions and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not specify any free parameters, axioms, or invented entities; the work focuses on algorithmic development and numerical validation based on standard quaternion algebra.

pith-pipeline@v0.9.0 · 5505 in / 1169 out tokens · 55714 ms · 2026-05-09T17:53:55.190586+00:00 · methodology

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Reference graph

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