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arxiv: 2606.08175 · v1 · pith:J5LD2FVQnew · submitted 2026-06-06 · 🧮 math.NA · cs.NA

Quaternion Maximum-Volume Submatrix Selection with Applications to Multichannel Imaging and Visual Data

Vsevolod Kliushev , Junjun Pan , Valentin Leplat This is my paper

Pith reviewed 2026-06-27 19:27 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords quaternion matricesmaximum-volume principleCUR approximationsubmatrix selectionmultichannel imagingquaternion least squareslow-rank approximation
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The pith

Maximum-volume submatrix selection extends to quaternion matrices with proofs that swaps increase volume.

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

The paper defines a maximum-volume principle for quaternion matrices by combining the Study determinant with quaternion singular values to choose rows and columns. It constructs two algorithms: one that greedily replaces rows and columns in a square core, and another that enlarges a row set while controlling interpolation coefficients. The work proves that accepted swaps raise the quaternion volume when the exact inverse is used, links the stopping rule to quasi-dominance, establishes an exact CUR identity for full-rank matrices, and supplies an interpolation stability bound. These results matter for multichannel data such as color images and motion capture because the non-commutative setting otherwise lacks a direct analogue of the classical volume criterion that guarantees stable low-rank approximations.

Core claim

The central claim is that the classical maximum-volume idea carries over to quaternions once volume is measured by the Study determinant of a submatrix together with its quaternion singular values; under this measure, row-and-column replacement algorithms can be built whose successful swaps strictly increase volume, the rectangular enlargement yields controlled coefficients, and the selected submatrices satisfy an exact CUR identity when the matrix has full rank.

What carries the argument

The quaternion maximum-volume criterion, which selects square or rectangular submatrices by maximizing a volume formed from the Study determinant and the quaternion singular values of the candidate block.

If this is right

  • Successful row and column swaps increase the quaternion volume of the selected square core when the exact quaternion inverse is used.
  • The stopping criterion of the algorithms is equivalent to a quasi-dominance condition on the interpolation matrix.
  • An exact quaternion CUR identity holds whenever the original matrix has full rank.
  • The rectangular procedure supplies an append-row pseudoinverse update that acts as a right preconditioner for overdetermined quaternion least-squares problems.

Where Pith is reading between the lines

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

  • The same volume-based selection logic could be tested on octonion or other non-commutative algebras that appear in higher-dimensional sensor data.
  • The preconditioning property might be examined on quaternion systems whose conditioning arises from geometric constraints rather than random noise.
  • Row selection in motion-capture data could be combined with existing skeleton-reduction pipelines to measure downstream reconstruction error.

Load-bearing premise

That a volume defined via the Study determinant and quaternion singular values remains a reliable proxy for interpolation stability and approximation accuracy in the presence of non-commutative multiplication.

What would settle it

Execute the square-core swap procedure on a concrete full-rank quaternion matrix and check whether any accepted swap decreases the computed volume or whether the resulting CUR factors fail to reproduce the original matrix exactly.

Figures

Figures reproduced from arXiv: 2606.08175 by Junjun Pan, Valentin Leplat, Vsevolod Kliushev.

Figure 1
Figure 1. Figure 1: Example of RGB reconstruction with quaternion Greedy MaxVol using Newton–Schulz [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of quaternion CUR decomposition for high resolution [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction demonstration for high resolution [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of RGB reconstruction with Tensor-CUR. [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between Greedy MaxVol and Tensor-CUR performance on Happywhale [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstruction demonstration of Greedy MaxVol and Tensor-CUR for high resolution [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between Greedy MaxVol and Tensor-CUR for high resolution [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Synthetic least-squares metrics for noise level [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Selected motion-change frames for RectMaxVol and the baseline methods on one [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: MoCap reconstruction error and setup time over [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Percentage of BVH records where each method gives the best reconstruction error. [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Paired differences between each baseline reconstruction error and the RectMaxVol [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Percentage of BVH records where RectMaxVol gives a smaller reconstruction error [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
read the original abstract

Low-rank approximation based on selected rows and columns is a useful alternative to singular value decompositions when the goal is an interpretable and compact matrix representation. A standard way to choose these rows and columns is the maximum-volume principle: it selects submatrices with large volume, which usually leads to stable interpolation coefficients and accurate CUR-type approximations. In this paper, we study this idea for quaternion matrices. This setting is natural for color images, three-dimensional motion data, and multi-channel signals, but requires care because quaternion multiplication is noncommutative. We define quaternion maximum-volume submatrix selection using quaternion singular values and the Study determinant. We then derive quaternion rank-one update formulas and use them to build two selection procedures: a greedy square-core method for row and column replacement, and a rectangular method that enlarges a selected row set until the interpolation coefficients are controlled. We prove that successful row and column swaps increase the quaternion volume of the selected square core when the exact quaternion inverse is used. We also connect the stopping criterion with quasi-dominance, prove an exact quaternion CUR identity in the full-rank case, and derive an interpolation stability bound. For the rectangular case, we derive an append-row pseudoinverse update and show how it gives a natural right preconditioner for overdetermined quaternion least-squares problems. Finally, we illustrate the methods on three applications: quaternion CUR approximation of RGB images, RectMaxVol-based preconditioning for ill-conditioned quaternion least-squares systems, and row selection in quaternion motion-capture data. The experiments show that the proposed quaternion MaxVol and RectMaxVol methods provide stable and efficient selection routines.

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 paper extends the maximum-volume (MaxVol) principle to quaternion matrices by defining volume via the Study determinant of quaternion singular values. It derives quaternion rank-one update formulas to implement greedy square-core row/column replacement and a rectangular enlargement procedure, proves that successful swaps increase volume when the exact inverse is used, connects the stopping criterion to quasi-dominance, establishes an exact CUR identity in the full-rank case, derives an interpolation stability bound, and presents an append-row pseudoinverse update that yields a right preconditioner for overdetermined quaternion least-squares problems. The methods are illustrated on RGB-image CUR approximation, ill-conditioned quaternion least-squares preconditioning, and row selection in motion-capture data.

Significance. If the algebraic claims hold, the work supplies an interpretable, SVD-free low-rank tool for non-commutative data arising in color imaging and 3-D motion capture. The explicit rank-one updates, CUR identity, and stability bound are concrete strengths that could be reused beyond the presented applications.

major comments (2)
  1. [Section deriving the greedy square-core method and the volume-increase proof] The central claim that successful row/column swaps strictly increase quaternion volume (when the exact inverse is used) rests on the Study determinant inheriting the required monotonicity under the derived quaternion rank-one updates. Because quaternion multiplication is non-commutative, the determinant identities that guarantee volume increase over reals or complexes do not transfer automatically; the manuscript must supply an explicit verification step for the Study determinant under these updates rather than invoking the real-valued case.
  2. [Definition of quaternion volume and the stability-bound derivation] The definition of quaternion volume (Study determinant of the matrix of quaternion singular values) is invoked both to define the selection criterion and to prove the stability bound. No edge-case verification is supplied for rank-deficient or near-singular quaternion matrices, where the non-commutative singular-value ordering could affect the claimed monotonicity or the quasi-dominance stopping criterion.
minor comments (2)
  1. [Experimental results on RGB images and motion-capture data] The experimental section would be strengthened by reporting the numerical condition numbers of the selected submatrices alongside the approximation errors, to directly illustrate the claimed stability advantage.
  2. [Rectangular append-row procedure] Notation for the quaternion pseudoinverse update in the rectangular case should be cross-referenced to the earlier square-core formulas to clarify reuse of the rank-one machinery.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the potential utility of the quaternion extensions. We address each major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

  1. Referee: [Section deriving the greedy square-core method and the volume-increase proof] The central claim that successful row/column swaps strictly increase quaternion volume (when the exact inverse is used) rests on the Study determinant inheriting the required monotonicity under the derived quaternion rank-one updates. Because quaternion multiplication is non-commutative, the determinant identities that guarantee volume increase over reals or complexes do not transfer automatically; the manuscript must supply an explicit verification step for the Study determinant under these updates rather than invoking the real-valued case.

    Authors: We agree that non-commutativity requires an explicit verification rather than direct appeal to the real or complex case. The manuscript derives the quaternion rank-one updates and asserts the volume increase for successful swaps when the exact inverse is used, but we acknowledge that the monotonicity argument for the Study determinant should be spelled out in detail. In the revision we will insert a dedicated lemma that computes the change in the Study determinant under the derived quaternion rank-one update formulas, confirming the strict increase whenever the swap condition is met. revision: yes

  2. Referee: [Definition of quaternion volume and the stability-bound derivation] The definition of quaternion volume (Study determinant of the matrix of quaternion singular values) is invoked both to define the selection criterion and to prove the stability bound. No edge-case verification is supplied for rank-deficient or near-singular quaternion matrices, where the non-commutative singular-value ordering could affect the claimed monotonicity or the quasi-dominance stopping criterion.

    Authors: We thank the referee for this observation. The volume is defined to be zero whenever the matrix is rank-deficient, and the selection procedures operate only on invertible square cores; nevertheless, we agree that near-singular and rank-deficient edge cases deserve explicit remarks. In the revised manuscript we will add a short discussion clarifying that (i) the volume is identically zero for rank-deficient matrices, (ii) monotonicity under swaps continues to hold for any invertible core, and (iii) the quasi-dominance stopping criterion remains well-defined provided the selected submatrix stays nonsingular. These remarks will be placed near the definition of quaternion volume and the stability bound. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions and proofs rest on external quaternion algebra and prior MaxVol framework

full rationale

The paper defines quaternion volume via the Study determinant applied to quaternion singular values, derives rank-one update formulas, and proves that swaps increase volume under exact inverse. These steps invoke standard non-commutative algebra and the established real-valued maximum-volume principle rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. The derivation chain is therefore self-contained against external benchmarks and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard quaternion algebra (non-commutativity, existence of inverses when defined) and on the prior real-valued MaxVol theory. No free parameters or new postulated entities are introduced in the abstract.

axioms (1)
  • domain assumption Quaternion multiplication is non-commutative and the Study determinant provides a suitable scalar volume measure for quaternion matrices.
    Invoked when defining the selection criterion and when proving volume increase under swaps.

pith-pipeline@v0.9.1-grok · 5833 in / 1445 out tokens · 16647 ms · 2026-06-27T19:27:27.199354+00:00 · methodology

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