Spectral Bounds for Tensors Derived from Trace Functionals and Wasserstein Distance in Tensor Spaces

Hemant Sharma; Nachiketa Mishra

arxiv: 2605.16930 · v1 · pith:VNDWSTCInew · submitted 2026-05-16 · 🧮 math.NA · cs.NA

Spectral Bounds for Tensors Derived from Trace Functionals and Wasserstein Distance in Tensor Spaces

Hemant Sharma , Nachiketa Mishra This is my paper

Pith reviewed 2026-05-19 19:49 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords positive semi-definite tensorsBures-Wasserstein distancetrace functionalseigenvalue boundstensor spacesspectral analysis

0 comments

The pith

Trace functionals produce eigenvalue bounds for positive semi-definite tensors while defining their Bures-Wasserstein distance.

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

The paper derives bounds on the eigenvalues of positive semi-definite tensors using trace functionals. It also introduces the Bures-Wasserstein distance as a metric on the space of these tensors. These constructions rely on the positive semi-definite property, and the paper examines what changes when the property is relaxed through examples. A complexity analysis of the associated computational methods completes the study.

Core claim

For positive semi-definite tensors, trace functionals yield eigenvalue bounds that depend on the semi-definiteness condition, and the same functionals induce the Bures-Wasserstein distance between pairs of such tensors.

What carries the argument

The Bures-Wasserstein distance on the space of positive semi-definite tensors, constructed from trace functionals that also deliver spectral bounds.

If this is right

Eigenvalue bounds are obtained directly from trace computations for any PSD tensor.
A geometric distance is available to compare different PSD tensors.
The bounds lose their guarantee when the positive semi-definite condition is removed.
The proposed methods have explicitly analyzed computational complexity.

Where Pith is reading between the lines

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

The framework could support new algorithms for tensor optimization problems that respect the PSD constraint.
Similar constructions might apply to other structured tensor classes beyond the PSD case.
Numerical implementations could test the bounds on concrete high-order tensor examples from applications.

Load-bearing premise

Tensors are assumed to be positive semi-definite so that the trace functionals produce the claimed bounds and metric properties.

What would settle it

Finding a specific tensor that is not positive semi-definite for which the proposed trace-based eigenvalue bound does not hold.

read the original abstract

This article introduces a trace-based metric on the space of positive semi-definite (PSD) tensors, offering a geometric perspective that connects their algebraic structure to their intrinsic geometric properties. It defines the Bures-Wasserstein distance on tensor spaces, establishing clear measurements between tensors. Moreover, the study derives trace-based eigenvalue bounds for PSD tensors and analyzes how these bounds depend on the PSD condition. The behavior of these bounds is further explored when the PSD requirement is relaxed, with illustrative examples provided to support the theoretical findings. In addition, a detailed complexity analysis is carried out for the methods proposed in this study.

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 sets up a Bures-Wasserstein metric on PSD tensors via trace functionals and derives eigenvalue bounds under that assumption, with relaxed cases shown only in examples.

read the letter

The key point is that this work defines the Bures-Wasserstein distance on PSD tensor spaces using trace functionals and derives eigenvalue bounds that rely on the PSD condition, with the non-PSD case handled through examples. They do a solid job connecting the trace to the metric properties and providing a complexity analysis that could help with implementation. The examples illustrate how the bounds shift when eigenvalues can be negative, which at least tests the dependence on the assumption. The limitation is that the main results stay conditional on positive semi-definiteness, and there is no general extension or proof for the relaxed setting. This could be a real constraint depending on the application. It is also not obvious from the abstract how much of this overlaps with prior work on tensor metrics or Wasserstein distances in other contexts. This paper is for people in numerical analysis or tensor geometry who might use these bounds in specialized methods. It is narrow enough that it will not appeal to everyone, but the derivations look grounded enough to send out for peer review. A referee could help sharpen the novelty claims and the handling of the PSD restriction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a trace-based metric on the space of positive semi-definite (PSD) tensors by defining the Bures-Wasserstein distance on tensor spaces. It derives trace-based eigenvalue bounds for PSD tensors, analyzes their dependence on the PSD condition, explores the behavior when the PSD requirement is relaxed via illustrative examples, and includes a complexity analysis of the proposed methods.

Significance. If the derivations hold, the work provides a geometric perspective linking the algebraic structure of PSD tensors to intrinsic properties via the Bures-Wasserstein distance and trace functionals. This could be useful in numerical analysis applications involving tensors. Credit is given for the explicit complexity analysis, which supports practical assessment, and for including examples to illustrate the relaxed PSD case.

major comments (2)

Abstract and core sections on derivations: The eigenvalue bounds and metric properties (non-negativity, symmetry, triangle inequality) are obtained under the positive semi-definite assumption. The abstract states that the relaxed PSD case is explored only through illustrative examples rather than by extending the main derivations or proving modified bounds. This renders the central claims conditional on a hypothesis that does not always hold in tensor applications, as noted in the stress-test concern.
Section on relaxed PSD analysis (presumably §5): The illustrative examples for non-PSD tensors should include quantitative assessment of bound tightness or explicit counterexamples where the trace functional fails to satisfy the claimed inequalities, to evaluate whether the headline results remain valid beyond the PSD setting.

minor comments (2)

Notation and definitions: The trace functional and induced distance could benefit from more explicit early definitions and consistent notation to improve readability for readers in numerical analysis.
References: Ensure all relevant prior work on Bures-Wasserstein metrics in matrix spaces and tensor spectral theory is cited to properly contextualize the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below in a point-by-point manner, indicating where revisions will be made to improve clarity and strengthen the presentation of results.

read point-by-point responses

Referee: Abstract and core sections on derivations: The eigenvalue bounds and metric properties (non-negativity, symmetry, triangle inequality) are obtained under the positive semi-definite assumption. The abstract states that the relaxed PSD case is explored only through illustrative examples rather than by extending the main derivations or proving modified bounds. This renders the central claims conditional on a hypothesis that does not always hold in tensor applications, as noted in the stress-test concern.

Authors: We appreciate the referee's observation. The eigenvalue bounds and metric properties are rigorously derived under the PSD assumption, as stated in the core sections. The abstract correctly describes the relaxed PSD case as being explored through illustrative examples rather than modified proofs, without extending the main claims. To address any potential ambiguity about the conditional nature of the results, we will revise the abstract to more explicitly state that the primary theoretical contributions require the PSD condition, while the examples illustrate the effects of relaxing it. This is a clarification that does not change the manuscript's scope or focus. revision: yes
Referee: Section on relaxed PSD analysis (presumably §5): The illustrative examples for non-PSD tensors should include quantitative assessment of bound tightness or explicit counterexamples where the trace functional fails to satisfy the claimed inequalities, to evaluate whether the headline results remain valid beyond the PSD setting.

Authors: We agree that quantitative assessments would enhance the examples. In the revised version, we will add numerical evaluations of bound tightness for the existing examples and include explicit counterexamples showing cases where the trace functional inequalities fail without the PSD condition. This will better demonstrate the role of the PSD assumption. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivations rest on explicit PSD assumptions and standard trace definitions

full rationale

The paper's core claims involve defining a trace-based Bures-Wasserstein metric on PSD tensors and deriving eigenvalue bounds that explicitly depend on the positive semi-definite condition. These steps are presented as direct consequences of the trace functional and the PSD hypothesis rather than reductions to fitted parameters, self-citations, or tautological redefinitions. The relaxed-PSD case is addressed only through separate illustrative examples, which does not affect the independence of the main derivations. No load-bearing step reduces by construction to its own inputs, and the abstract provides no indication of ansatz smuggling or uniqueness theorems imported from prior self-work. The analysis is therefore self-contained against external benchmarks for the stated domain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions from linear algebra and metric geometry on tensors; no free parameters or invented entities are apparent from the abstract.

axioms (1)

domain assumption Tensors under consideration are positive semi-definite
Invoked as the condition under which the metric and eigenvalue bounds are derived and analyzed.

pith-pipeline@v0.9.0 · 5626 in / 925 out tokens · 39832 ms · 2026-05-19T19:49:49.092879+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 3.4: For PSD Hermitian tensors A,B, n∑ λi(A)λn−i+1(B) ≤ tr(A∗B) ≤ n∑ λi(A)λi(B).
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

Definition of d(A,B) via tr(A)+tr(B)−2 tr((A^{1/2}∗B∗A^{1/2})^{1/2})^{1/2} on PSD cone.

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.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Symmetric decomposition of the associ- ated graded algebra of an Artinian Gorenstein algebra

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Lund, The tensor T-function: a definition for functions of third-order tensors, Numer

K. Lund, The tensor T-function: a definition for functions of third-order tensors, Numer. Linear Algebra Appl. (2020).https://doi.org/10.1002/nla.2288

work page doi:10.1002/nla.2288 2020

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Factorization strategies for third-order tensors

Kilmer, Misha E., and Carla D. Martin. "Factorization strategies for third-order tensors." Linear Algebra and its Applications 435, no. 3 (2011): 641-658.https://doi.org/10.1016/j.laa.2010.09.020

work page doi:10.1016/j.laa.2010.09.020 2011

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Numerical optimization for symmetric tensor decomposition

Kolda, Tamara G. "Numerical optimization for symmetric tensor decomposition." Mathematical Programming 151 (2015): 225-248.https://doi.org/10.1007/s10107-015-0895-0

work page doi:10.1007/s10107-015-0895-0 2015

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Tensor decompositions and applications

Kolda, Tamara G., and Brett W. Bader. "Tensor decompositions and applications." SIAM review 51, no. 3 (2009): 455-500.https://doi.org/10.1137/07070111X

work page doi:10.1137/07070111x 2009

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Eigenvalue bounds of third-order tensors via the minimax eigenvalue of sym- metric matrices

Li, Shigui, et al. "Eigenvalue bounds of third-order tensors via the minimax eigenvalue of sym- metric matrices." Computational and Applied Mathematics 39.3 (2020): 1-14. https://doi.org/10.1007/ s40314-020-01245-0

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Liu, Y.-S

Luo, Shunlong, and Qiang Zhang. "Informational distance on quantum-state space." Physical Review A—Atomic, Molecular, and Optical Physics 69, no. 3 (2004): 032106.https://doi.org/10.1103/PhysRevA. 69.032106

work page doi:10.1103/physreva 2004

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The Fréchet derivative of the tensor T-function

Lund, Kathryn, and Marcel Schweitzer. "The Fréchet derivative of the tensor T-function." Calcolo 60, no. 3 (2023): 35. https://doi.org/10.1007/s10092-023-00527-3

work page doi:10.1007/s10092-023-00527-3 2023

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Pusz, Wieslaw, and S. Lech Woronowicz. "Functional calculus for sesquilinear forms and the purifi- cation map." Reports on Mathematical Physics 8, no. 2 (1975): 159-170.https://doi.org/10.1016/ 19 20 H. Sharma and N. Mishra 0034-4877(75)90061-0 1Department of Mathematics Indian Institute of Information Technology, Design and Manufacturing Kancheepuram, Ch...

work page 1975