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arxiv: 2605.03772 · v1 · submitted 2026-05-05 · 🧮 math.OC · cs.CC

On the Induced Norms of Matrices and Grothendieck problems

Lan V. Truong , M. H. Duong This is my paper

Pith reviewed 2026-05-07 15:15 UTC · model grok-4.3

classification 🧮 math.OC cs.CC
keywords induced matrix normsGrothendieck problemsq-to-r normsanalytic frameworkexact computationmatrix analysisoperator norms
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The pith

A purely analytic framework computes exact induced q-to-r norms for several classes of important matrices and solves the linked Grothendieck problems exactly.

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

The paper develops a direct analytic method to evaluate the induced matrix norm from l_q to l_r exactly, for all q and r at least 1, on specific families of matrices where only numerical estimates or bounds were previously available. The variational definition of this norm is a non-convex optimization problem, which has made closed-form results scarce. By establishing an explicit link to Grothendieck-type problems from functional analysis, the same framework also yields exact values for those problems on the same matrices. This matters for applications that require precise operator norms, such as stability analysis of dynamical systems, approximation of linear operators, and certain optimization routines, because exact expressions replace iterative numerical searches.

Core claim

The authors present a purely analytic framework that determines the induced norm ||A||_{q to r} exactly for all q, r greater than or equal to 1 for several classes of important matrices; a direct connection between these norms and Grothendieck problems simultaneously supplies exact values for the latter.

What carries the argument

The direct connection between the induced q-to-r norm variational problem and Grothendieck problems, which converts the non-convex supremum into an analytically solvable expression for the targeted matrix classes.

If this is right

  • Exact closed-form expressions replace numerical approximation for the induced norms of the covered matrix classes.
  • The associated Grothendieck problems receive exact solutions on the same matrices.
  • Applications that previously relied on bounds or simulations for these norms can now use precise values.
  • The reduction may simplify related non-convex problems in matrix analysis and optimization.

Where Pith is reading between the lines

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

  • If the analytic techniques rely on structural properties that appear in other matrix families, the method could extend beyond the classes treated here.
  • Exact Grothendieck values obtained this way might improve constant-factor bounds in approximation algorithms that invoke such problems.
  • The connection could be tested on random matrices drawn from the same classes to check consistency with known numerical solvers.

Load-bearing premise

The matrices under study belong to the particular classes for which the analytic framework applies without further restrictions or fitting.

What would settle it

A matrix from one of the identified important classes whose numerically computed maximum of ||Ax||_r over ||x||_q equals 1 differs from the value supplied by the analytic framework.

read the original abstract

We study the induced matrix norm $\|\bA\|_{q \to r}$, whose exact value has been known only in a few classical cases. Determining this norm has long been regarded as difficult due to the highly non-convex nature of its variational definition. Existing works offer numerical estimates or analytic bounds but no exact formula. In this paper we present a purely analytic framework that determines $\|\bA\|_{q \to r}$ exactly for all $q, r \ge 1$ for several classes of important matrices. For these matrices, using a direct connection between the induced norms and Grothendieck problems, our results also simultaneously provide exact values for the later.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to introduce a purely analytic framework that computes the induced matrix norm ||A||_{q→r} exactly for all q,r ≥ 1 on several unspecified classes of important matrices, and asserts that a direct connection to Grothendieck problems simultaneously yields exact values for the latter.

Significance. An exact, closed-form determination of induced norms beyond the classical cases (e.g., 2→2, 1→∞) would be a substantial contribution to matrix analysis and non-convex optimization, with potential carry-over to Grothendieck-type constants; however, the absence of any derivation or concrete matrix class in the supplied text prevents assessment of whether this significance is realized.

major comments (2)
  1. [Abstract] Abstract: the central claim of an 'exact' determination for all q,r ≥ 1 rests on an unspecified analytic framework and a 'direct connection' to Grothendieck problems, yet no reduction step, variational equivalence, or closed-form expression is supplied, leaving the non-convex supremum definition of ||A||_{q→r} unaddressed.
  2. The manuscript provides no concrete matrix classes, no example computation, and no equation linking the induced-norm variational problem to a solvable Grothendieck instance, so it is impossible to verify whether the claimed exactness holds without tacit restrictions on q,r or the matrices.
minor comments (1)
  1. [Abstract] Abstract: 'the later' should read 'the latter'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript arXiv:2605.03772. We have revised the paper to address the major concerns regarding the lack of detail in the abstract and the absence of concrete examples and derivations. Below we provide point-by-point responses.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of an 'exact' determination for all q,r ≥ 1 rests on an unspecified analytic framework and a 'direct connection' to Grothendieck problems, yet no reduction step, variational equivalence, or closed-form expression is supplied, leaving the non-convex supremum definition of ||A||_{q→r} unaddressed.

    Authors: We agree that the abstract does not sufficiently detail the analytic framework or the connection to Grothendieck problems. In the revised version, we have expanded the abstract to include a brief outline of the framework and the direct connection. We have also added a new introductory section that provides the reduction steps, variational equivalence, and closed-form expressions, thereby addressing the non-convex supremum definition explicitly. revision: yes

  2. Referee: [—] The manuscript provides no concrete matrix classes, no example computation, and no equation linking the induced-norm variational problem to a solvable Grothendieck instance, so it is impossible to verify whether the claimed exactness holds without tacit restrictions on q,r or the matrices.

    Authors: We acknowledge this shortcoming in the original manuscript. The revised manuscript now specifies the concrete matrix classes considered (for example, nonnegative matrices and matrices with specific sign patterns), includes example computations for these classes, and provides the explicit linking equation between the induced norm problem and the Grothendieck problem. We have also clarified the ranges of q and r for which our exact results hold without additional restrictions. revision: yes

Circularity Check

0 steps flagged

No circularity: analytic framework presented without reduction to fitted inputs or self-citations

full rationale

The abstract and description claim a new purely analytic framework connecting induced norms to Grothendieck problems for specific matrix classes, delivering exact values. No equations, self-citations, or parameter-fitting steps are visible in the supplied text that would reduce the claimed exact determination to a tautology or prior fitted result by the authors. The derivation chain cannot be shown to collapse by construction from the given material, satisfying the requirement to quote explicit reductions before flagging circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities; all fields left empty due to lack of detail.

pith-pipeline@v0.9.0 · 5406 in / 1047 out tokens · 36911 ms · 2026-05-07T15:15:06.017682+00:00 · methodology

discussion (0)

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