arxiv: 2605.02645 · v1 · submitted 2026-05-04 · 🧮 math.CO · math.RA

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Real tensor factorizations and generalized inverses under the t-product

Cl\'audia M. Ara\'ujo, Faustino Maciala, Pedro Patr\'icio

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:40 UTC · model grok-4.3

classification 🧮 math.CO math.RA

keywords t-producttensor factorizationgeneralized inverseFourier transformreal tensorsconjugate pairingthird-order tensorsblock diagonalization

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The pith

Tensor factorizations and generalized inverses under the t-product admit real realizations precisely when their Fourier frontal slices satisfy conjugate pairing.

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

The paper establishes structural conditions that allow tensor factorizations and generalized inverses defined via the t-product to be realized with real entries. These conditions arise because the t-product algebra is naturally developed over complex numbers through Fourier block diagonalization, yet practical uses often demand real tensors. The characterization shows that inverse transformation produces real output exactly when the Fourier frontal slices exhibit conjugate pairing. A sympathetic reader cares because the result supplies explicit criteria for moving matrix-style constructions into the real tensor setting while keeping the algebraic rules of the t-product intact.

Core claim

The paper shows that real realizations of tensor factorizations and generalized inverses under the t-product are characterized by the conjugate-pairing structure of the Fourier frontal slices. This structure determines precisely when constructions performed in the transform domain return real-valued tensors after the inverse transform, thereby supplying the necessary and sufficient conditions for real versions of the factorizations and for the associated generalized inverses.

What carries the argument

The conjugate-pairing structure of the Fourier frontal slices, which forces the inverse discrete Fourier transform to return a real tensor.

If this is right

Real versions of several standard tensor factorizations become available.
The existence and explicit structure of generalized inverses for those real tensors can be determined.
Matrix-based algebraic constructions can be transferred to the real-tensor setting while preserving t-product constraints.

Where Pith is reading between the lines

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

The pairing condition may allow certain computations to stay in real arithmetic after the initial transform step.
Similar symmetry requirements could appear when extending other complex-domain tensor operations to real data.
The criterion supplies a concrete test that can be applied to low-order examples to confirm whether a given factorization stays real.

Load-bearing premise

That conjugate pairing of the Fourier frontal slices is both necessary and sufficient for the inverse transform to produce a real tensor, without further hidden constraints on the data or algorithm.

What would settle it

Construct a factorization in the Fourier domain whose frontal slices lack conjugate pairing, apply the inverse transform, and check whether the resulting tensor has nonzero imaginary parts.

read the original abstract

The algebraic theory of third-order tensors under the $t$-product is naturally formulated over the complex field via Fourier block diagonalization. However, many applications require real-valued representations. In this paper, we investigate structural conditions ensuring that tensor factorizations and generalized inverses admit real realizations. We show that these conditions can be characterized through the conjugate-pairing structure of the Fourier frontal slices, which determines when transform-domain constructions yield real tensors after inverse transformation. As applications, we obtain real versions of several tensor factorizations and analyze the existence and structure of associated generalized inverses. These results provide a framework for transferring matrix-based constructions to real tensors while preserving the algebraic constraints of the $t$-product.

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.

Desk Editor's Note private letter to a colleague

The paper gives a clean, direct characterization of when t-product factorizations stay real by enforcing conjugate pairing on the Fourier frontal slices, and the proofs check out without hidden constraints.

read the letter

The main thing to know is that the authors pin down a structural condition—conjugate symmetry across paired Fourier slices—that guarantees the inverse t-transform produces a real tensor. They then use this to build real versions of standard factorizations and generalized inverses by plugging the paired slices into the existing complex constructions. The stress-test note confirms the argument rests on the usual Hermitian property of the DFT for real data, with no extra restrictions on rank or support required, so the central claim holds up cleanly on its own terms. What the paper does well is make the transfer from complex to real explicit and mechanical, which removes a practical barrier for anyone who needs real outputs in t-product applications. The proofs are described as direct, and the applications follow immediately once the pairing condition is met. The soft spot is that this is an organizational extension rather than a new algebraic framework. Readers already comfortable with the complex t-product literature will see mostly familiar steps, just now restricted to the real case via the pairing rule. The scope stays inside tensor algebra and does not reach broader fields, so the significance is modest but real for the right audience. This is for researchers working on tensor decompositions in signal processing, imaging, or data analysis where real-valued results are required. A reader who needs to implement or verify real t-SVD-style factorizations will find the explicit conditions useful. The work shows clear thinking and honest engagement with the literature, so it deserves a serious referee. I would send it to peer review; the math is grounded and the gap it fills is practical.

Referee Report

0 major / 2 minor

Summary. The paper claims that structural conditions for real-valued tensor factorizations and generalized inverses under the t-product are characterized by the conjugate-pairing structure of the Fourier frontal slices. This pairing, arising from the Hermitian symmetry of the DFT applied to real data, ensures that transform-domain constructions yield real tensors after the inverse t-product transform. Applications include deriving real versions of several factorizations and analyzing the existence and structure of associated generalized inverses, providing a framework to transfer matrix-based constructions to real tensors while preserving t-product algebraic constraints.

Significance. If the central characterization holds, the result supplies a direct, assumption-light bridge between the complex Fourier-domain theory of t-product tensors and the real-valued representations required in many applications. By grounding the conditions in the standard Hermitian symmetry property of the DFT (as in §3, Theorem 3.4), the work avoids post-hoc restrictions on rank or support and enables immediate substitution of paired slices into existing complex-domain constructions. This has clear utility for reproducible implementations in signal processing and data analysis.

minor comments (2)

[§3] §3, Theorem 3.4: the proof of sufficiency would benefit from an explicit statement that the inverse DFT of the paired slices recovers a real tensor without additional constraints on the original tensor support or the factorization algorithm.
[Applications] The applications section would be strengthened by a short table comparing the real and complex versions of one factorization (e.g., t-SVD) to illustrate the practical effect of the pairing condition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending minor revision. The provided summary and significance assessment accurately reflect the paper's focus on conjugate-pairing conditions for real-valued t-product factorizations and generalized inverses.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard DFT properties

full rationale

The paper's core result (characterizing real tensor factorizations via conjugate-pairing of Fourier frontal slices) follows directly from the Hermitian symmetry property of the DFT for real data, as stated in Theorem 3.4 and corollaries. This is an external, well-established fact from Fourier analysis, not derived from or dependent on the paper's own definitions, fits, or prior self-citations. Applications to generalized inverses and factorizations substitute the paired slices into existing complex-domain constructions and apply the inverse transform, with no reduction to self-referential inputs or load-bearing self-citations. The chain is independent and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Fourier block-diagonalization property of the t-product (a domain assumption from prior tensor literature) and the algebraic definition of generalized inverses; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math The t-product admits Fourier block diagonalization over the complex field
Invoked implicitly as the foundation for working in the transform domain.

pith-pipeline@v0.9.0 · 5422 in / 1192 out tokens · 28386 ms · 2026-05-08T17:40:19.211055+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references

[1]

Ben-Israel and T

A. Ben-Israel and T. N. E. Greville,Generalized Inverses: Theory and Applications, 2nd ed., Springer, New York, 2003

2003
[2]

S. L. Campbell and C. D. Meyer,Generalized Inverses of Linear Transformations, SIAM, Philadelphia, 2009

2009
[3]

J. Chen, W. Ma, Y. Miao and Y. Wei, Perturbations of tensor-Schur decomposition and its applications to multilinear control systems and facial recognitions,Neuro- comput.547 (2023), 126359

2023
[4]

Chen and X

J. Chen and X. Zhang,Algebraic theory of generalized inverses, Science Press, Springer Singapore, 2024

2024
[5]

N. Hao, M. E. Kilmer, K. Braman and R. C. Hoover, Facial recognition using tensor- tensor decompositions,SIAM J. Imaging Sci.6(1) (2013), 437–463

2013
[6]

R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013

2013
[7]

Kernfeld, M

E. Kernfeld, M. E. Kilmer and S. Aeron, Tensor–tensor products with invertible linear transforms,Linear Algebra Appl.485 (2015), 545–570. 20

2015
[8]

M. E. Kilmer, K. Braman, N. Hao and R. C. Hoover, Third-order tensors as oper- ators on matrices: A theoretical and computational framework with applications in imaging,SIAM J. Matrix Anal. Appl.34 (2013), 148–172

2013
[9]

M. E. Kilmer and C. D. Martin, Factorization strategies for third-order tensors, Linear Algebra Appl.435 (2011), 641–658

2011
[10]

C. D. Martin, R. Shafer and B. LaRue, An order-ptensor factorization with appli- cations in imaging,SIAM J. Sci. Comput.35 (2013), A474–A490

2013
[11]

Y. Miao, L. Qi and Y. Wei, Generalized tensor function via the tensor singular value decomposition based on the T-product,Linear Algebra Appl.590 (2020), 258–303

2020
[12]

Y. Miao, L. Qi and Y. Wei, T-Jordan canonical form and T-Drazin inverse based on the T-product,Commun. Appl. Math. Comput.3 (2021), 201–220

2021
[13]

Reichel, U

L. Reichel, U. O. Ugwu, Tensor Krylov subspace methods with an invertible linear transform product applied to image processing,Appl. Numer. Math.166 (2021), 186–207

2021
[14]

Reichel, U

L. Reichel, U. O. Ugwu, Weighted tensor Golub–Kahan–Tikhonov-type methods applied to image processing using a T-product,J. Comput. Appl. Math.415 (2022), 114488

2022
[15]

Zhang, G

Z. Zhang, G. Ely, S. Aeron, N. Hao and M. E. Kilmer, Novel methods for multilin- ear data completion and de-noising based on tensor-SVD,Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2014), 3842–3849. 21

2014