Biquadratic Tensors: Eigenvalues and Structured Tensors

Chunfeng Cui; Liqun Qi

REVIEW 4 major objections 6 minor 1 cited by

Biquadratic Tensors: Eigenvalues and Structured Tensors

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Pith's one-line read A biquadratic tensor is positive semi-definite exactly when every one of its M-eigenvalues is nonnegative.

desk verdict Correct variational characterization of PSD for nonsymmetric biquadratic tensors, but the eigenvalue framing does less work than advertised and several secondary proofs need patching. read the letter →

arxiv 2502.06447 v2 pith:767IJXS2 submitted 2025-02-10 math.SP

Liqun Qi , Chunfeng Cui This is my paper · ORCID

classification math.SP MSC 47J1015A1847H0715A72

keywords BiquadratictensorsM-eigenvaluespositivesemi-definitenessGershgorin-typetheoremdiagonallydominatedM-biquadraticB-biquadraticRiemannianLBFGS

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Biquadratic tensors are fourth-order arrays $A=(a_{i_1j_1i_2j_2})$ paired with the quartic form $f(x,y)=\sum a_{i_1j_1i_2j_2}x_{i_1}y_{j_1}x_{i_2}y_{j_2}$; covariance tensors, elasticity tensors, and the Riemann curvature tensor are all biquadratic yet only weakly symmetric. The paper extends M-eigenvalues to these nonsymmetric tensors and proves that a biquadratic tensor is positive semi-definite if and only if all its M-eigenvalues are nonnegative, and positive definite if and only if all its M-eigenvalues are positive. Since each M-eigenvalue equals the value of $f$ at its eigenvectors, the smallest M-eigenvalue is exactly the global minimum of $f$ on the product of two unit spheres. The paper also provides a Gershgorin-type interval theorem, introduces structured classes that certify positivity from simple entry inequalities, and gives a Riemannian LBFGS algorithm for computing the smallest M-eigenvalue. Positivity certification for these applied tensors is thereby reduced to a single eigenvalue computation.

What carries the argument

The carrying mechanism is the constrained minimization of the biquadratic form $f(x,y)=\langle A,x\circ y\circ x\circ y\rangle$ on the compact product of two unit spheres $x^\top x=1$, $y^\top y=1$. A global minimizer exists by continuity, and its first-order optimality conditions are precisely the equations defining an M-eigenpair, with the multiplier identity $\lambda=f(x,y)$; hence the smallest M-eigenvalue is the global minimum. Around this core, the Gershgorin-type theorem localizes all M-eigenvalues in intervals built from diagonal entries $a_{ijij}$ and dominance radii $r_{ij}$, the structured classes (diagonally dominated, Z-, M-, strong M-, B0-, and B-biquadratic tensors) turn simple entry inequalities into positivity certificates, and the Riemannian LBFGS algorithm minimizes $f$ through Cayley-transform retractions on the two spheres, with convergence resting on a sufficient-decrease bound and the Kurdyka-Lojasiewicz property of the semi-algebraic objective.

What would settle it

The identity $\lambda=f(x,y)$ at stationary points can be tested directly: generate a random nonsymmetric biquadratic tensor, compute the global minimum of $f$ on the product of unit spheres by exhaustive sampling or certified global optimization, and compare it with the smallest $\lambda$ satisfying the M-eigenvalue equations. If the sampled minimum ever lies strictly below that eigenvalue, Theorem 3.1 is false. For the algorithm, a simpler check is whether the total path length $\sum_k\|z^{(k+1)}-z^{(k)}\|$ of the generated iterates is finite in practice, since Theorem 6.6 asserts global convergence.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.1: every real biquadratic tensor has at least one M-eigenvalue, and the signs of all M-eigenvalues completely decide positive semi-definiteness and definiteness. An M-eigenvalue is a real $\lambda$ for which there are unit vectors $x\in\mathbb{R}^m$ and $y\in\mathbb{R}^n$ satisfying the two stationarity equations (5) and (6), and multiplying those equations by $x$ and $y$ gives $\lambda=\langle A,x\circ y\circ x\circ y\rangle$. This identity makes the smallest M-eigenvalue equal to the minimum of the biquadratic form on the product of two unit spheres. The paper therefore reduces positivity of general, not necessarily symmetric, biquadratic tensors to checking the smallest M-eigenvalue, and it supplies verifiable sufficient conditions and a numerical method for that check.

Load-bearing premise

For the algorithmic convergence claim, the load-bearing premise is that the iterates produced by the safeguarded Riemannian LBFGS method satisfy the sufficient-decrease and gradient-comparison conditions required by the cited convergence theorem; if those conditions do not hold for optimization on this product of two spheres, the global convergence statement of Theorem 6.6 is not established, although the M-eigenvalue characterization of positivity stands independently.

Editorial extensions

If this is right

Every weakly symmetric biquadratic tensor arising as a covariance tensor, elasticity tensor, or Riemann curvature tensor can have its positive semi-definiteness decided by the sign of its smallest M-eigenvalue, even though these tensors are not symmetric in general.
The Gershgorin-type theorem gives explicit intervals containing all M-eigenvalues; when each diagonal entry $a_{ijij}$ dominates the corresponding radius $r_{ij}$, Corollary 4.2 certifies positive semi-definiteness or definiteness without computing any eigenvalue.
M-biquadratic tensors and symmetric B0-biquadratic tensors are always positive semi-definite, and strong M-biquadratic tensors and symmetric B-biquadratic tensors are always positive definite; these are checkable entrywise classes.
Under the paper's convergence assumptions, the Riemannian LBFGS algorithm produces iterates that converge to an M-eigenpair and whose gradient norm tends to zero; the numerical experiments report recovery of the smallest M-eigenvalue in repeated trials, with success rate declining as dimensions grow.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension (not in the paper): because the smallest M-eigenvalue is the global minimum of the biquadratic form, the characterization ties directly to polynomial-optimization and sum-of-squares methods, so weakly symmetric tensors from statistics and mechanics could be certified by degree-four SOS certificates even when symmetric-tensor criteria do not apply.
Extension (not in the paper): the Gershgorin intervals give a cheap pre-screening test for any iterative solver: if the strict diagonal-dominance inequalities already hold, positivity is certified without computing eigenvalues; the testable question is how often these intervals certify positivity for random weakly symmetric covariance and elasticity tensors of moderate dimension.
Extension (not in the paper): Example 3.3 shows that even diagonal biquadratic tensors can have additional M-eigenvalues beyond their diagonal entries, so in higher dimensions generic tensors likely have many M-eigenpairs; this would make the observed decline in success rates of global optimization a structural feature rather than a solver deficiency.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main theorem is correct: for a nonsymmetric biquadratic tensor, PSD/PD is equivalent to nonnegativity/positivity of its M-eigenvalues, where M-eigenvalues are KKT points of the same quadratic form on the product of two unit spheres. The proof is a short KKT argument and it works. The genuinely new pieces are the extension of M-eigenvalues to nonsymmetric tensors, a Gershgorin-type inclusion theorem, and structured tensor classes (M-, B-, diagonally dominated) with checkable sufficient conditions. The Gershgorin bounds are the most useful part: they give explicit intervals that can certify PSD/PD without solving an optimization problem, and the comparisons to prior bounds are fair.

That said, the central theorem is close to a tautology. Since M-eigenvalues are defined as stationary points of f itself, saying "all M-eigenvalues are nonnegative" is essentially saying "the global minimum of f is nonnegative." The paper does not hide this, but the framing as a powerful tool oversells it. The real value sits in the sufficient conditions.

Several secondary results are under-proved. Proposition 3.4 claims extra M-eigenvalues of a diagonal tensor lie in the convex hull of diagonal entries, but the proof only derives a condition involving the convex hull of the rows and columns; it does not establish the stated claim. Theorem 5.5's decomposition uses index sets J_k that are not maintained as Cartesian products, so the tensors I_J as defined are not controlled properly; the induction looks fixable but needs real work. Corollary 5.4 cites [11, Thm 7] for invariance of M-eigenvalues under diagonal scaling, but that theorem concerns Tucker decompositions and does not say what is claimed; the corollary can be recovered by a direct scaling argument, so the correct response is to fix the citation and proof, not to drop the result.

The convergence analysis of the Riemannian LBFGS method invokes the Kurdyka-\L{}ojasiewicz framework of Bolte-Sabach-Teboulle without verifying the required exponent or the retraction-specific inequalities on the product of spheres. As written, the global convergence of iterates in Theorem 6.6 is not proven. This does not affect Theorem 3.1, but the algorithm section's headline claim overreaches. The numerical section reports no code or data, so the experiments are not reproducible from the paper itself.

Who will get value: people working on strong ellipticity or tensor eigenvalue bounds will want the Gershgorin theorem and the structured tensor definitions. The paper deserves a serious referee: the main theorem is correct, the Gershgorin theorem is a real contribution, and the gaps are patchable rather than fatal. I would send it to review with instructions to the referee to focus on the incomplete proofs and the KL step. After a revision that fixes Proposition 3.4, Theorem 5.5, the citation in Corollary 5.4, and either proves or appropriately weakens Theorem 6.6, it would be a solid paper.

Referee Report

4 major / 6 minor

Summary. The paper extends the theory of M-eigenvalues from symmetric to general (nonsymmetric) biquadratic tensors of dimension m x n x m x n. It proves that every biquadratic tensor has an M-eigenvalue and that a tensor is positive semidefinite (positive definite) if and only if all of its M-eigenvalues are nonnegative (positive), via the KKT conditions of the associated minimization over two unit spheres. It then gives a Gershgorin-type inclusion interval for M-eigenvalues, derives positive semidefiniteness for (strictly) diagonally dominated tensors, introduces Z-, M-, strong M-, B0-, and B-biquadratic tensors, and claims PSD/PD results for these classes. Finally, it presents a safeguarded Riemannian LBFGS algorithm for computing the smallest M-eigenvalue, with a convergence analysis and numerical experiments on elasticity, covariance, and random tensors.

Significance. If the main characterization holds, it provides a clean and elementary bridge between a variational definition and eigenvalue-based PSD/PD checking for nonsymmetric biquadratic tensors, which is useful for applications in elasticity, statistics, and relativity. The Gershgorin-type bounds are concrete and appear to give competitive intervals in the reported examples. The structured-tensor results extend an established line of work from cubic tensors to the biquadratic setting. The central Theorem 3.1 is proved correctly and does not rely on any unstated parameters or ad-hoc assumptions. The secondary results are less polished: the proof of Theorem 5.5 has a structural gap, and the convergence proof of Algorithm 1 invokes a KL framework without fully verifying its hypotheses. These issues do not undermine Theorem 3.1 but do affect the completeness of the paper's claims about structured tensors and global convergence.

major comments (4)

[§5, Theorem 5.5 and definition of I_J] The decomposition proof is not valid as written. The theorem uses tensors I_J in (22), defined only for J = (Jx, Jy) with I_J(i1,j1,i2,j2)=1 if i1,i2 ∈ Jx and j1,j2 ∈ Jy. However, the proof defines J1 as a set of pairs (i,j), namely {(i,j) : ∃(i2,j2)≠(i,j) such that b_{ij i2 j2} > 0}, and then subtracts h1 I_J1. The Cartesian product structure required for I_J1 is not present: even if the pairs in J1 are viewed as a relation, subtracting h1 on the full product of the x- and y-coordinates of J1 affects entries whose index pairs are not in J1, and the argument that B2 remains a B0-biquadratic tensor does not control those entries. The claimed inclusion J_{k+1} = J_k \ \hat{J}_k also does not resolve this mismatch. Since Corollary 5.6 relies on Theorem 5.5, the PSD claim for symmetric B0-biquadratic tensors lacks a correct proof in the current text.
[§6.1, Theorem 6.6] The global convergence proof invokes the Kurdyka-Lojasiewicz framework of Bolte-Sabach-Teboulle [1] without verifying that all hypotheses hold for the product-of-two-unit-spheres problem with the Cayley retraction (30)-(31). The sufficient decrease and gradient lower bound conditions are claimed to follow from Lemmas 6.3-6.5, but those lemmas quote Lemma 4.3 and Lemma 4.7 of [3], which are stated for a single unit sphere. The extension to the product manifold, the simultaneous Armijo backtracking over one scalar α, and the precise KL property for the rational objective restricted to the manifold are not demonstrated. Thus the statement that the sequence z(k) converges globally to an M-eigenvector is not proven. This is a real incompleteness in the numerical section, although it does not affect Theorem 3.1.
[§5, Corollary 5.4] The proof appeals to 'Theorem 7 in [11]' for the claim that an M-eigenvalue of C := A ×1 D ×2 F ×3 D ×4 F is also an M-eigenvalue of A. This is not correct: positive diagonal scaling does not preserve M-eigenvalues in general, and the cited invariance statement for Tucker cores does not apply in this form. The corollary's conclusion is nonetheless true and can be recovered directly from ⟨C, x∘y∘x∘y⟩ = ⟨A, (Dx)∘(Fy)∘(Dx)∘(Fy)⟩, since D and F are positive diagonal matrices. The proof should be replaced with this direct argument.
[§3, Proposition 3.4] The proof does not establish the stated claim that all M-eigenvalues of a diagonal biquadratic tensor lie in the convex hull of some diagonal entries. The displayed argument only gives a conditional existence statement depending on whether the all-ones vectors lie in certain convex hulls; it never shows that an arbitrary M-eigenpair (λ, x, y) must satisfy that condition. A short direct proof is available: from (11), for each i with x_i ≠ 0, λ = Σ_j a_{ijij} y_j^2, which is a convex combination of entries of the i-th row of the diagonal matrix D. The proposition should be re-proven with this argument or the proof should be amended to conclude exactly what is claimed.

minor comments (6)

[§1, first paragraph] There is a spacing typo: 'Let m, nbe positive integers' should read 'Let m, n be positive integers'.
[§2.3] The phrase 'the space of allvector fieldson M' is missing spaces; it should read 'the space of all vector fields on M'.
[§5, Proposition 5.1, equation (20)] The displayed formula for the Z-biquadratic case has missing absolute-value bars and unbalanced parentheses: the terms involving |¯a_{i1jij2}| and |¯a_{i1j1ij}| are not correctly separated. The right-hand side should be a_{ijij} - (1/4)( Σ_{i1} Σ_{j2} |¯a_{i1jij2}| + Σ_{i1} Σ_{j1} |¯a_{i1j1ij}| + Σ_{i2} Σ_{j2} |¯a_{iji2j2}| + Σ_{i2} Σ_{j1} |¯a_{ij1i2j}| ).
[§5, Theorem 5.5] The theorem statement uses 'symmetric B0-quadratic tensor' and 'symmetric B-quadratic tensor' where the established terminology in the paper is 'B0-biquadratic' and 'B-biquadratic'. This should be made consistent.
[§5, Proposition 5.3] The proof labels the two directions as 'necessity' and 'sufficiency' in a way that is opposite to the conventional usage: the first paragraph invokes Proposition 5.2 for the direction M-biquadratic ⇒ PSD, while the construction of B = γI − A proves PSD ⇒ M-biquadratic. The logic is correct, but the labels should be swapped or the exposition clarified.
[§8] There is a typo in the final remarks: 'a biquadrtic tensor' should be 'a biquadratic tensor'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central M-eigenvalue/PSD theorem is a proved variational equivalence, and the self-citations appearing in the convergence analysis are independent prior mathematical results, not fitted inputs or definitional substitutes.

full rationale

The paper's central claim, Theorem 3.1, states that a biquadratic tensor is positive semi-definite (definite) if and only if all of its M-eigenvalues are nonnegative (positive). M-eigenvalues are introduced as the KKT points of the same quadratic form f(x,y) whose nonnegativity defines PSD, so the theorem is indeed a direct variational characterization: the global minimizer of f on the compact product of unit spheres is an M-eigenpair with λ = f(x,y), and every M-eigenpair has λ equal to the value of f at that point. This makes the equivalence immediate, but it is proved from the definitions rather than assumed, and it is not an instance of fitting a parameter and then renaming it as a prediction. No fitted values, hidden parameters, or data-derived inputs are used anywhere in the derivation. The self-citations that appear are not load-bearing in a circular way: the original M-eigenvalue paper [21] is a historical reference for the symmetric case, the structured-tensor book [23] supplies background definitions and a template for Theorem 5.5 whose proof is then given in the paper, and the Cayley-transform lemmas [3] used in the convergence analysis are prior published results with independent mathematical content rather than assertions of the present paper's conclusions. The possible incompleteness in Theorem 6.6, where the Kurdyka-Lojasiewicz framework of [1] is invoked without fully checking all retraction-specific hypotheses, and the questionable appeal to [11, Thm 7] in Corollary 5.4, are correctness or completeness concerns, not circularity: they do not reduce the paper's conclusions to their own assumptions by construction. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central theorems are parameter-free. Algorithm hyperparameters in Section 7 are implementation choices and are not fitted to data. The derivations rely on standard compactness and KKT arguments plus one semi-algebraic and KL assumption for the algorithm's global convergence.

assumptions (3)

standard math The product of unit spheres in R^m and R^n is compact and f is continuous, so a global minimizer exists.
Used in Theorem 3.1 to prove existence of M-eigenvalues.
standard math The constraints x^Tx=1 and y^Ty=1 satisfy the linear independence constraint qualification, so KKT conditions hold at the global minimizer and at other stationary points.
Used in Theorem 3.1 and in Section 6 to identify KKT points with M-eigenpairs.
domain assumption The objective and constraints in problem (24) are semi-algebraic and satisfy the Kurdyka-Lojasiewicz property, and Theorem 1 of Bolte-Sabach-Teboulle [1] applies to the safeguarded Riemannian LBFGS sequence.
Invoked in Theorem 6.6 for global convergence of iterates; the hypotheses are cited rather than verified in detail.

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Cite this review

Pith. "Pith review of Biquadratic Tensors: Eigenvalues and Structured Tensors." pith.science (2026). https://pith.science/paper/767IJXS2

@misc{pith2026250206447,
  author       = {Pith},
  title        = {Pith review of: Biquadratic Tensors: Eigenvalues and Structured Tensors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/767IJXS2}},
  note         = {Machine review of arXiv:2502.06447}
}

abstract

The covariance tensors in statistics{, elasticity tensor in solid mechanics, Riemann curvature tensor in relativity theory are all biquadratic tensors that are weakly symmetric, but not symmetric in general. Motivated by this, in this paper, we consider nonsymmetric biquadratic tensors, and study possible conditions and algorithms for identifying positive semi-definiteness and definiteness of such biquadratic tensors. We extend M-eigenvalues to nonsymmetric biquadratic tensors, prove that a general biquadratic tensor has at least one M-eigenvalue, and show that a general biquadratic tensor is positive semi-definite if and only if all of its M-eigenvalues are nonnegative, and a general biquadratic tensor is positive definite if and only if all of its M-eigenvalues are positive. We present a Gershgorin-type theorem for biquadratic tensors, and show that (strictly) diagonally dominated biquadratic tensors are positive semi-definite (definite). We introduce Z-biquadratic tensors, M-biquadratic tensors, strong M-biquadratic tensors, B$_0$-biquadratic tensors and B-biquadratic tensors. We show that M-biquadratic tensors and symmetric B$_0$-biquadratic tensors are positive semi-definite, and that strong M-biquadratic tensors and symmetric B-biquadratic tensors are positive definite. A Riemannian LBFGS method for computing the smallest M-eigenvalue of a general biquadratic tensor is presented. Numerical results are reported.

Figures

Figures reproduced from arXiv: 2502.06447 by the authors.

**Figure 2.** The iterative procedure of Algorithm 1 for computing M-eigenvalues of a nonsym [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗

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L.M. Zubov and A.N. Rudev, “On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials”, Journal of Applied Mathematics and Mechanics 9 (2016) 1096-1102. 31

2016

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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Proximal alternating linearized minimizationfor nonconvex and nonsmooth problems

J. Bolte, S. Sabach and M. Teboulle, “Proximal alternating linearized minimizationfor nonconvex and nonsmooth problems”, Mathematical Programming, Series A 146 (2014) 459494

work page 2014

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Chang, Y

J. Chang, Y. Chen and L. Qi, “Computing eigenvalues of large scale sparse tensors arising from a hypergraph, SIAM Journal on Scientific Computing 38 (6) (2016) A3618-A3643

work page 2016

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Tensor and its tucker core: The invariance relationships

B. Jiang, F. Yang and S. Zhang, “Tensor and its tucker core: The invariance relationships”, Numerical Linear Algebra with Applications 24 (3) (2017) No. e2086

work page 2017

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Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

I.M. Bomze, C. Ling, L. Qi and X. Zhang, “Standard bi-quadratic optimization problems and unconstrained polynomial reformulations”, Journal of Global Optimization, 52 (2012) 663-687

work page 2012

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New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors

H. Che, H. Chen and G. Zhou, “New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors”, Journal of Industrial and Manage- ment Optimization 17(6) (2021) 3685-3694

work page 2021

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Block structure-based covariance tensor decomposition for group identification in matrix variables

Y. Chen, Z. Hu, J. Hu and L. Shu, “Block structure-based covariance tensor decomposition for group identification in matrix variables”, Statistics and Probability Letters 216 (2025) No. 110251

work page 2025

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Spectra of uniform hypergraphs

J. Cooper and A. Dutle, “Spectra of uniform hypergraphs”, Linear Algebra and Its Appli- cations 436 (2012) 3268-3292. 28

work page 2012

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Elasticity M-tensors and the strong ellipticity condi- tion

W. Ding, J. Liu, L. Qi and H. Yan, “Elasticity M-tensors and the strong ellipticity condi- tion”, Applied Mathematics and Applications 373 (2020) No. 124982

work page 2020