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Sharp H-eigenvalue bounds for positive definite tensors come from exact Lagrangian extremization over power-sum and determinant invariants, not AM-GM.

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2026-07-10 12:51 UTC pith:2J4KMZHP

load-bearing objection Clean, honest extension of the authors’ own AM–GM tensor bounds: exact Lagrangian solutions plus a K-cluster structural theorem that actually tightens the envelopes.

arxiv 2607.08113 v1 pith:2J4KMZHP submitted 2026-07-09 math.OC cs.NAmath.NA

Sharp Spectral Bounds for Symmetric Positive Definite Tensors via Multiple Algebraic Invariants

Hemant Sharma , Snigdhashree Nayak , Ankit Singh This is my paper
classification math.OC cs.NAmath.NA MSC 15A1815A6915A4265F1593D05
keywords H-eigenvalue boundssymmetric tensorspower sumsLagrangian extremizationLyapunov stabilityregion of attractiontrace-determinant bounds
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows how to get sharp upper and lower bounds on the H-eigenvalues of a symmetric positive definite tensor by treating the spectrum as the solution of a constrained optimization problem whose constraints are algebraic invariants of the tensor (trace, higher power sums, and determinant). Earlier work used the AM-GM inequality as a convenient but loose surrogate; here the exact stationarity conditions are solved, and the resulting bounds are attained on admissible cluster spectra. A structural theorem then proves that any maximizer constrained by K such invariants can take at most K distinct values, so the search collapses to a finite list of low-dimensional polynomial systems. The resulting hierarchy of bounds tightens monotonically toward the true spectral radius. On random spectra the three-invariant bound already cuts the median overestimation gap from 53 percent to 6 percent, remains cheap up to dimension 100, and enlarges certified Lyapunov regions of attraction by factors of two to three.

Core claim

Any maximizer of the largest eigenvalue over the K-invariant feasibility region has at most K distinct spectral values; consequently the sharp upper bound is the largest root of a finite collection of low-dimensional polynomial systems, and the hierarchy of those bounds is monotonically tightening and is attained precisely when the spectrum itself has at most K clusters.

What carries the argument

The K-invariant structural theorem (any extremizer over F_K has at most K distinct values) together with the explicit two-invariant polynomial φ_{T,D}(Λ)=Λ(T-Λ)^{d-1}-D(d-1)^{d-1} whose largest real root is the sharp two-invariant upper bound.

Load-bearing premise

The whole argument needs every H-eigenvalue to be real and positive so they can be ordered and treated as ordinary positive numbers; that fails for a generic higher-order positive definite tensor that carries complex-conjugate eigenvalues.

What would settle it

Take a symmetric positive definite tensor whose H-spectrum is known to be real and has five or more distinct values; compute the four invariants and run the four-invariant solver; if the returned upper bound equals the true largest eigenvalue rather than strictly exceeding it, the sharpness claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper extends the authors' prior AM–GM trace–determinant bounds for H-eigenvalues of symmetric positive definite tensors by replacing the AM–GM relaxation with exact Lagrangian extremization over the spectral feasibility region F_K defined by K algebraic invariants (power sums and the determinant). The central structural result (Theorem 5.1) is that any maximizer of λ_1 over F_K has at most K distinct values, so the sharp upper bound B^{+}_K is obtained by solving a finite union of low-dimensional polynomial systems; this yields the monotone hierarchy B^{+}_2 ≥ B^{+}_3 ≥ B^{+}_4 ≥ ⋯ ≥ λ_max, with B^{+}_2 given explicitly as the largest root of the univariate polynomial φ_{T,D}. The four-invariant case is developed in detail (Bézout-type counts, multistart Newton algorithm, sharpness theorem), closed forms are given for d ≤ 4, and the framework is applied to Lyapunov ROA estimates and validated on random spectra up to d = 100 and on genuine tensors with real H-spectrum.

Significance. If the results hold, the paper supplies a clean, algebraically sharp hierarchy of H-eigenvalue bounds that strictly dominate the authors' earlier AM–GM bounds and recover classical matrix inequalities (Merikoski–Virtanen, Wolkowicz–Styan) as special cases. The structural theorem is elementary but useful: it reduces an infinite-dimensional spectral optimization to finitely many low-dimensional polynomial systems whose cost is independent of tensor order once the invariants are known. Strengths include explicit algorithms, a perturbation analysis showing fortunate insensitivity to the expensive determinant, reproducible numerical evidence that the three-invariant gap falls from ~53 % to ~6 % median, and a carefully delimited real-H-spectrum scope. The Lyapunov ROA application demonstrates a concrete payoff (2–3× larger certified regions). Within the stated class (matrices, even-order diagonal/orthogonally decomposable tensors) the contribution is solid and immediately usable.

minor comments (5)
  1. Section 12.4 correctly states the real-H-spectrum hypothesis, but a short forward pointer in the introduction (or after Lemma 2.2) would help readers who might otherwise assume the bounds apply to generic even-order tensors.
  2. Algorithm 2 (multistart Newton) would benefit from a brief note on how N_starts and I_max were chosen in the experiments of §6.5–6.7, so that the reported exact recovery of λ_max = 1.1 is reproducible without trial-and-error.
  3. Table 2 reports microsecond timings for diagonal tensors; a one-sentence remark that these are dominated by language overhead rather than arithmetic would prevent misreading the scaling.
  4. A few typographical inconsistencies remain (e.g., “B´ ezout” vs. “Bézout”, occasional missing spaces around em-dashes). A final copy-edit pass would polish the presentation.
  5. Figure 1 caption could state the exact generator (Exp(2)+0.5) and seed policy already used in the text, so the figure is self-contained.

Circularity Check

0 steps flagged

No significant circularity: bounds follow from KKT stationarity and algebraic elimination on independently defined invariants.

full rationale

The paper's central results (Theorems 3.1, 4.1, 5.1, 6.1–6.4) are obtained by writing the Lagrangian for max λ₁ subject to power-sum and product constraints, reading off the stationarity polynomial of degree ≤K−1 for the non-outlier eigenvalues, and reducing to finite low-dimensional systems. The invariants T, S, p_k, D are defined independently of the bounds (via generalized traces and the resultant). Self-citation of Nayak–Sharma–Mishra [1] is only the AM–GM baseline being strictly improved (Corollary 3.2); it is not used as a uniqueness theorem or as an input that forces the new bounds. Numerical examples use independently chosen spectra or genuine tensors and recover classical matrix bounds (Merikoski–Virtanen, Wolkowicz–Styan) as special cases. Sharpness statements are if-and-only-if characterizations of when the spectrum lies in the optimizing cluster family, not tautologies that redefine the target. The real-positive H-spectrum hypothesis is an explicit scope restriction (§12.4), not a circular premise. No fitted parameter is renamed a prediction, and no load-bearing uniqueness is imported from the authors' prior work. Score 0 is therefore appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper is pure constrained optimization over spectral invariants. It imports standard Lagrange/KKT theory, Bézout finiteness, Newton identities, and the Hu–Huang–Ling–Qi trace formulas, plus the domain restriction to real positive H-spectra of SPD tensors. No numerical free parameters are fitted; the only modeling choice that limits applicability is the real-spectrum hypothesis. No new physical entities are postulated.

axioms (5)
  • standard math Karush–Kuhn–Tucker / Lagrange first-order conditions characterize interior maximizers of λ₁ over the compact positive feasibility set F_K (Lemma 2.2, Theorems 3.1, 4.1, 5.1).
    Classical constrained optimization; used throughout Sections 3–6.
  • standard math Bézout’s theorem bounds the number of isolated complex solutions of the four-equation cluster system by the product of degrees (Proposition 6.3).
    Standard algebraic geometry finiteness tool for the four-invariant solver.
  • domain assumption Power sums p_k and the determinant D of a symmetric tensor are computable from entries via generalized traces and the resultant (Qi; Hu–Huang–Ling–Qi).
    Background tensor algebra cited from [3,6]; required so that bounds are certificates from entries without solving the eigenproblem.
  • domain assumption The H-spectrum is real and positive, so eigenvalues admit the ordering λ₁≥⋯≥λ_d>0 used in every optimization (Section 12.4).
    Explicit scope restriction; automatic for matrices and even-order diagonal/OD tensors, not for generic m≥4 SPD tensors.
  • domain assumption A is symmetric positive definite of even order, so all H-eigenvalues are positive (Remark 1 of the prior paper).
    Needed for the positive orthant feasibility regions and log-determinant constraint.

pith-pipeline@v1.1.0-grok45 · 25667 in / 3047 out tokens · 31351 ms · 2026-07-10T12:51:08.905454+00:00 · methodology

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read the original abstract

We extend the trace--determinant framework of Nayak, Sharma, and Mishra~\cite{nayak2026} for bounding the H-eigenvalues of symmetric positive definite tensors. First, we replace the Arithmetic--Geometric Mean (AM--GM) relaxation underlying previous bounds by the exact solution of the associated constrained optimization problem, yielding sharp upper and lower bounds that are attained on the admissible spectral variety. Second, we incorporate higher-order power sums as additional spectral invariants and prove a structural theorem showing that any extremizer over a $K$-invariant feasibility region has at most $K$ distinct spectral values. This reduces the problem to a finite collection of low-dimensional polynomial systems and yields a hierarchy of increasingly tight bounds. For the four-invariant case $(T,S,p_3,D)$, we develop a complete theory including solution-count estimates, a multistart Newton algorithm, and sharpness conditions. We also derive closed-form bounds in small dimensions, establish perturbation estimates, and obtain refined Lyapunov region-of-attraction bounds. Numerical experiments for dimensions up to $d=100$ show that the sharp three-invariant bound reduces the median relative overestimation gap from $53\%$ to $6\%$ while maintaining low computational cost. The framework is validated on tensors with real H-spectrum.

Figures

Figures reproduced from arXiv: 2607.08113 by Ankit Singh, Hemant Sharma, Snigdhashree Nayak.

Figure 1
Figure 1. Figure 1: Distribution of relative overestimation gaps over 100 random spectra ( [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Runtime scaling of the sharp bounds (mean over 20 random spectra per [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Level sets of V (x) = 1.1 x 4 1 + x 4 2 (solid contours) and guaranteed inscribed ℓ4-balls inside {V ≤ 4} obtained from upper bounds on λmax (dashed curves). Tighter upper bounds yield strictly larger inscribed regions. The sharp 3-invariant bound (dark green) gives a region 52% larger in area than the AM–GM bound (red); the sharp 4-invariant bound (which would coincide with “True” here, dotted black) is e… view at source ↗

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Reference graph

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