Sufficient and Necessary Conditions for Eckart-Young like Result for Tubal Tensors

Uria Mor

arxiv: 2512.24405 · v3 · submitted 2025-12-30 · 🧮 math.NA · cs.NA

Sufficient and Necessary Conditions for Eckart-Young like Result for Tubal Tensors

Uria Mor This is my paper

Pith reviewed 2026-05-16 18:41 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords tubal tensorsEckart-Young theoremlow-rank approximationtubal singular value decompositiontensor algebraFrobenius norm

0 comments

The pith

Certain tubal products make the truncated tubal SVD the exact minimizer of Frobenius error among lower-rank tensor approximations.

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

The paper determines the precise algebraic conditions on a tubal product that make the Eckart-Young theorem hold for tensors. Under those conditions the best low-rank approximation in the Frobenius norm is obtained simply by discarding the smallest singular tubes of the tubal SVD. This gives a complete necessary-and-sufficient characterization rather than an existence result for one particular product. The authors then apply the result to video data and to data-driven dynamical systems to show that the identified products support practical low-rank modeling.

Core claim

We provide a complete characterization of the family of tubal products that yield an Eckart-Young type result, i.e., for which the truncation of the tubal SVD produces the minimal Frobenius-norm error among all tensors of strictly lower tubal rank.

What carries the argument

The tubal product together with its induced tubal SVD, whose truncation error is minimized precisely when the product satisfies the derived algebraic conditions.

If this is right

Low-rank tensor approximation reduces to a single SVD computation followed by truncation for every product in the characterized family.
Existing matrix-style algorithms for compression, denoising, and completion transfer directly once the right tubal product is chosen.
Video sequences and dynamical-system trajectories can be modeled with provably optimal low-rank truncations under these products.
Any tubal product violating the conditions admits at least one tensor where truncation is strictly suboptimal.

Where Pith is reading between the lines

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

The characterization supplies a design criterion for inventing new tubal products tailored to specific data modalities.
The same algebraic test may decide whether other matrix identities, such as orthogonality preservation, survive in the tensor setting.
Numerical stability of the truncated SVD can now be studied uniformly across the entire family rather than product by product.

Load-bearing premise

The tubal product must admit a well-defined SVD for which the approximation error after truncation can be expressed algebraically in terms of the singular tubes.

What would settle it

A concrete tubal product that satisfies the paper's algebraic conditions yet fails to make SVD truncation optimal on some explicit tensor, or a product outside the family that nevertheless succeeds on all tested tensors.

Figures

Figures reproduced from arXiv: 2512.24405 by Uria Mor.

**Figure 2.** Figure 2: Showing filtration results using TSVDM γ = .45 leading components w.r.t the DCT (middle column) and the scaled DCT (right), for video frames 20 and 100 in the testing dataset. Dynamic Mode Decomposition In our next numerical demonstration, we showcase the nonuniformly scaled transforms’ advantages when applied to the tensor based dynamic mode decomposition method from [28]. In short: given a tubal-tensor X… view at source ↗

**Figure 3.** Figure 3: DMD experiment results. Left: variation in DMD relative error ( [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

**Figure 4.** Figure 4: 2d state reconstruction for the last snapshot in [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗

**Figure 5.** Figure 5: The result of adding noise to the data and automatically masking moving objects [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗

**Figure 6.** Figure 6: Showing the resulting tSVDMII approximations using the DCT (middle column, one component) [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗

read the original abstract

A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Importantly, it has been shown that for a specific family of tubal products, an Eckart-Young type theorem holds, i.e., the best low-rank approximation of a tensor under the Frobenius norm is obtained by truncating its tubal SVD. In this paper, we provide a complete characterization of the family of tubal products that yield an Eckart-Young type result. We demonstrate the practical implications of our theoretical findings by conducting experiments with video data and data-driven dynamical systems.

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 necessary-and-sufficient characterization of tubal products that preserve the Eckart-Young property for Frobenius low-rank approximation.

read the letter

The central contribution is a full algebraic description of which tubal products make the truncated tubal SVD the optimal low-rank approximation in Frobenius norm. That is new; earlier papers had the result only for particular choices like the standard t-product. The authors prove sufficiency directly and necessity by exhibiting counterexample tensors for any product outside the identified family. The experiments on video data and data-driven dynamical systems are a reasonable way to show the practical difference those conditions make. The characterization itself looks algebraically grounded and the citation pattern is appropriate for the subfield. The main soft spot is the stress-test concern: the necessity direction appears to rely on the tubal product being realized through a fixed invertible transform (FFT or similar) that turns the problem into independent matrix multiplications. If a product is defined without that diagonalization structure or if the transform can be singular at some frequencies, the counterexamples may not apply and the claimed necessity could be narrower than stated. The paper should flag that assumption explicitly. Readers who work on tensor SVDs, video processing, or dynamical-system modeling will get direct value from the result. It is worth sending to referees because the claim is precise, the proof strategy is standard for this area, and the applications are concrete enough to test.

Referee Report

2 major / 2 minor

Summary. The paper claims to deliver a complete (sufficient and necessary) algebraic characterization of the family of tubal products for which the truncated tubal SVD yields the optimal low-rank approximation in the Frobenius norm, thereby extending the Eckart-Young theorem to the tubal-tensor setting. It further illustrates the result with experiments on video data and data-driven dynamical systems.

Significance. If the characterization is fully rigorous, the result would be a useful clarification of the precise conditions under which the Eckart-Young property holds for tubal products, strengthening the theoretical toolkit for tensor algebra in applications such as video processing and dynamical-system modeling. The paper's explicit necessity argument via counterexamples and its inclusion of practical validation are positive features.

major comments (2)

[§4] §4 (necessity direction of the main characterization theorem): the counterexample construction and the derivation of the necessary conditions both presuppose that the tubal product is realized by a fixed invertible linear transform (typically the FFT) that diagonalizes the product into independent matrix multiplications. It is not shown that the stated algebraic conditions remain necessary when the tubal product is defined abstractly without such a diagonalization or when the transform is singular at some frequencies.
[§5] §5 (experimental validation): the experiments are described only at a high level; it is unclear which concrete tubal products satisfying versus violating the derived conditions are compared, and whether the observed approximation errors directly corroborate the necessity claim.

minor comments (2)

Notation for the tubal product and the associated SVD should be introduced with an explicit list of standing assumptions (e.g., associativity, distributivity, existence of identity) before the main theorems.
The abstract states that experiments are conducted, yet the manuscript would benefit from a short reproducibility paragraph listing the precise tensor dimensions, chosen transforms, and error metrics used in §5.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and insightful review of our manuscript. The comments highlight important aspects regarding the scope of our theoretical results and the clarity of the experimental section. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

Referee: [§4] §4 (necessity direction of the main characterization theorem): the counterexample construction and the derivation of the necessary conditions both presuppose that the tubal product is realized by a fixed invertible linear transform (typically the FFT) that diagonalizes the product into independent matrix multiplications. It is not shown that the stated algebraic conditions remain necessary when the tubal product is defined abstractly without such a diagonalization or when the transform is singular at some frequencies.

Authors: We agree that the necessity argument in §4 is developed under the standard assumption that the tubal product admits an invertible linear transform (such as the FFT) that block-diagonalizes the operation into independent matrix multiplications. This is the setting in which the tubal tensor framework is most commonly defined and analyzed in the literature. For purely abstract tubal products lacking an explicit diagonalizing representation, the necessity claim would indeed require additional hypotheses; our characterization is therefore stated for the concrete, transform-based case. We will insert a short clarifying paragraph at the beginning of §4 and in the introduction to make this scope explicit and to note that invertibility of the transform is used to guarantee that the algebraic conditions are both necessary and sufficient. revision: partial
Referee: [§5] §5 (experimental validation): the experiments are described only at a high level; it is unclear which concrete tubal products satisfying versus violating the derived conditions are compared, and whether the observed approximation errors directly corroborate the necessity claim.

Authors: We accept this criticism. In the revised version we will expand §5 to name the specific tubal products employed (both those satisfying the derived algebraic conditions and those violating them), supply the precise parameter values used to construct the counterexamples, and tabulate the Frobenius-norm approximation errors of the truncated tubal SVD against the true optimal low-rank approximant. These quantitative comparisons will directly illustrate that the Eckart-Young property holds precisely when the conditions are met and fails otherwise. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic characterization is self-contained

full rationale

The paper derives sufficient and necessary algebraic conditions on tubal products for which truncated tubal SVD yields the optimal Frobenius low-rank approximation. Sufficiency follows from direct verification of the Eckart-Young property under the stated product axioms; necessity is shown by explicit counterexample tensors for products violating the conditions. No step reduces a claimed prediction to a fitted parameter, self-citation, or definitional tautology. The derivation relies on the abstract definition of the tubal product and its induced SVD, without importing uniqueness theorems or ansatzes from prior self-work that would force the result. The characterization therefore stands as an independent algebraic theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, new entities, or ad-hoc axioms are stated. The work relies on the pre-existing tubal-tensor framework and its SVD construction.

axioms (1)

domain assumption Tubal products admit an SVD-like decomposition whose truncation properties can be analyzed
Implicit in the statement that an Eckart-Young type result holds for a family of products.

pith-pipeline@v0.9.0 · 5397 in / 1040 out tokens · 31243 ms · 2026-05-16T18:41:20.992787+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 5.1: R_n satisfies Eckart-Young optimality for tubal-length truncation iff M = D Q with D = diag(μ_1 I_{d1}, …, μ_ℓ I_{dℓ}), Q unitary and rows constant on each ς(j).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.4 & Lemma 3.7: length(R_n) = ℓ via chain of principal ideals ⟨e_j⟩ ≅ F^{(j)}.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Copositive Matrices with Ordered Off-Diagonal Entries
math.OC 2026-05 unverdicted novelty 7.0

Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

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L. D. Lathauwer, B. D. Moor, and J. Vandewalle. A multilinear singular value decomposition.SIAM Journal on Matrix Analysis and Applications, 21(4):1253–1278, Jan. 2000

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E. Newman and K. Keegan. Optimal matrix-mimetic tensor algebras via variable projection, 2024

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A. Sobral and A. Vacavant. A comprehensive review of background subtraction algorithms evaluated with synthetic and real videos.Computer Vision and Image Understanding, 122:4–21, May 2014

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L. R. Tucker. Some mathematical notes on three-mode factor analysis.Psychometrika, 31(3):279–311, Sept. 1966. 28 A Rings, Ideals, and Modules A ring is a setRequipped with two binary operations, addition + and multiplication·, such that (R,+) is an Abelian group, multiplication is associative, distributive over addition from both sides, and there exists n...

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[1] [1]

Ahmed, T

N. Ahmed, T. Natarajan, and K. R. Rao. Discrete cosine transform.IEEE transactions on Computers, 100(1):90–93, 2006

work page 2006

[2] [2]

Altman and S

A. Altman and S. Kleiman.A Term of Commutative Algebra. Worldwide Center of Mathematics, 2012

work page 2012

[3] [3]

Avron and U

H. Avron and U. Mor. Demystifying tubal tensor algebra, 2025

work page 2025

[4] [4]

Ballard and T

G. Ballard and T. G. Kolda.Tensor decompositions for data science. Cambridge University Press, Cambridge, England, June 2025

work page 2025

[5] [5]

Bouwmans

T. Bouwmans. Traditional and recent approaches in background modeling for foreground detection: An overview.Computer Science Review, 11–12:31–66, May 2014

work page 2014

[6] [6]

K. Braman. Third-Order Tensors as Linear Operators on a Space of Matrices.Linear Algebra and its Applications, 433(7):1241–1253, 12 2010

work page 2010

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S. L. Brunton and J. N. Kutz.Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, May 2022

work page 2022

[8] [8]

Brutzer, B

S. Brutzer, B. H¨ oferlin, and G. Heidemann. Evaluation of background subtraction techniques for video surveillance. InCVPR 2011, pages 1937–1944. IEEE, 2011

work page 2011

[9] [9]

De Lathauwer, B

L. De Lathauwer, B. De Moor, and J. Vandewalle. On the best rank-1 and rank-(r1 ,r2 ,. . .,rn) approx- imation of higher-order tensors.SIAM Journal on Matrix Analysis and Applications, 21(4):1324–1342, 2000. 27

work page 2000

[10] [10]

G. H. Golub and C. F. Van Loan.Matrix Computations - 4th Edition. Johns Hopkins University Press, Philadelphia, PA, 2013

work page 2013

[11] [11]

Gutchess, M

D. Gutchess, M. Trajkovics, E. Cohen-Solal, D. Lyons, and A. K. Jain. A background model ini- tialization algorithm for video surveillance. InProceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001, volume 1, pages 733–740. IEEE, 2001

work page 2001

[12] [12]

explana- tory

R. a. Harshman. Foundations of the PARAFAC procedure: Models and conditions for an “explana- tory” multimodal factor analysis.UCLA Working Papers in Phonetics, 16(10), 1970

work page 1970

[13] [13]

C. J. Hillar and L.-H. Lim. Most tensor problems are NP-hard.Journal of the ACM, 60(6):1–39, Nov. 2013

work page 2013

[14] [14]

F. L. Hitchcock. The expression of a tensor or a polyadic as a sum of products.Journal of Mathematics and Physics, 6(1–4):164–189, Apr. 1927

work page 1927

[15] [15]

M. H. Holmes.Introduction to Scientific Computing and Data Analysis. Springer International Publishing, 2023

work page 2023

[16] [16]

Keegan and E

K. Keegan and E. Newman. Projected tensor-tensor products for efficient computation of optimal multiway data representations.Linear Algebra and its Applications, 729:100–147, Jan. 2026

work page 2026

[17] [17]

Kernfeld, M

E. Kernfeld, M. Kilmer, and S. Aeron. Tensor–tensor products with invertible linear transforms. Linear Algebra and its Applications, 485:545–570, Nov. 2015

work page 2015

[18] [18]

M. E. Kilmer, K. Braman, N. Hao, and R. C. Hoover. Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging.SIAM Journal on Matrix Analysis and Applications, 34(1):148–172, 2 2013

work page 2013

[19] [19]

M. E. Kilmer, L. Horesh, H. Avron, and E. Newman. Tensor-tensor algebra for optimal repre- sentation and compression of multiway data.Proceedings of the National Academy of Sciences, 118(28):e2015851118, July 2021

work page 2021

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M. E. Kilmer and C. D. Martin. Factorization strategies for third-order tensors.Linear Algebra and its Applications, 435(3):641–658, 8 2011

work page 2011

[21] [21]

M. E. Kilmer, C. D. Martin, and L. Perrone. A third-order generalization of the matrix svd as a product of third-order tensors, 2008

work page 2008

[22] [22]

T. G. Kolda and B. W. Bader. Tensor decompositions and applications.SIAM Review, 51(3):455–500, Aug. 2009

work page 2009

[23] [23]

K. Koor, Y. Qiu, L. C. Kwek, and P. Rebentrost. A short tutorial on wirtinger calculus with applications in quantum information.arXiv preprint arXiv:2312.04858, 2023

work page arXiv 2023

[24] [24]

J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor.Dynamic Mode Decomposition: Data- Driven Modeling of Complex Systems. Society for Industrial and Applied Mathematics, Nov. 2016

work page 2016

[25] [25]

L. D. Lathauwer, B. D. Moor, and J. Vandewalle. A multilinear singular value decomposition.SIAM Journal on Matrix Analysis and Applications, 21(4):1253–1278, Jan. 2000

work page 2000

[26] [26]

Newman and K

E. Newman and K. Keegan. Optimal matrix-mimetic tensor algebras via variable projection, 2024

work page 2024

[27] [27]

I. V. Oseledets. Tensor-train decomposition.SIAM Journal on Scientific Computing, 33(5):2295– 2317, 2011

work page 2011

[28] [28]

A. K. Saibaba, M. E. Kilmer, K. Hall-Hooper, F. Tian, and A. Mize. A tensor-based dynamic mode decomposition based on the⋆ m-product, 2025

work page 2025

[29] [29]

Sobral and A

A. Sobral and A. Vacavant. A comprehensive review of background subtraction algorithms evaluated with synthetic and real videos.Computer Vision and Image Understanding, 122:4–21, May 2014

work page 2014

[30] [30]

L. R. Tucker. Some mathematical notes on three-mode factor analysis.Psychometrika, 31(3):279–311, Sept. 1966. 28 A Rings, Ideals, and Modules A ring is a setRequipped with two binary operations, addition + and multiplication·, such that (R,+) is an Abelian group, multiplication is associative, distributive over addition from both sides, and there exists n...

work page 1966