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arxiv: 2605.08101 · v1 · submitted 2026-04-23 · 🧮 math.OC

Determinant Bounds for (n-1)-Locally Positive Semidefinite Matrices

Shaun Fallat , Samir Mondal , Hristo Sendov This is my paper

Pith reviewed 2026-05-12 01:17 UTC · model grok-4.3

classification 🧮 math.OC
keywords determinant boundspositive semidefinite matriceslocal positive semidefinitenessFisher inequalityKoteljanskii inequalityprincipal submatricesmatrix inequalities
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The pith

Sharp lower bounds on determinants quantify how far (n-1)-locally positive semidefinite matrices are from being globally positive semidefinite.

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

The paper studies matrices in which every proper principal submatrix is positive semidefinite, yet the full matrix satisfies only a determinant condition rather than full positive semidefiniteness. It derives sharp lower bounds on the determinants of these matrices to measure the precise gap between the local and global conditions. The same approach yields tight lower-bound versions of the classical Fisher and Koteljanskii determinant inequalities for this class. A reader cares because the bounds turn an abstract hierarchy of matrix properties into concrete, computable numbers that describe how close a matrix is to satisfying the stronger global requirement.

Core claim

For an (n-1)-locally positive semidefinite matrix the determinant admits a sharp lower bound strictly weaker than the non-negativity required by global positive semidefiniteness; the same technique supplies analogous sharp lower bounds that extend the Fisher and Koteljanskii inequalities to the local setting, thereby quantifying how far the local class can deviate from the global one.

What carries the argument

The (n-1)-locally positive semidefinite condition, meaning every proper principal submatrix is positive semidefinite while the full matrix is constrained only by its determinant, is the central object; the derived determinant inequalities quantify the deviation from global positive semidefiniteness.

If this is right

  • The new determinant lower bounds are attained by certain extremal matrices, confirming sharpness.
  • Fisher-type inequalities now supply lower bounds on products of principal minors for these locally constrained matrices.
  • Koteljanskii-type inequalities likewise receive tight lower bounds under the (n-1)-local condition.
  • The bounds give a quantitative distance from the local class to the global positive-semidefinite cone.

Where Pith is reading between the lines

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

  • The same locality-to-determinant technique could be tested at other levels of the principal-submatrix hierarchy.
  • The explicit bounds might serve as cheap certificates in numerical semidefinite-programming relaxations that stop short of full positive-semidefiniteness.
  • Connections between these determinant gaps and eigenvalue interlacing or inertia laws remain open for further exploration.

Load-bearing premise

Every proper principal submatrix is positive semidefinite while the full matrix satisfies only a determinant constraint.

What would settle it

An explicit (n-1)-locally positive semidefinite matrix whose determinant lies below the claimed sharp lower bound would falsify the result.

read the original abstract

In this framework, the extremal case corresponds to the tightest nontrivial relaxation in this hierarchy, in which every proper principal submatrix is constrained to be positive semidefinite, while the global positive semidefiniteness condition is governed by the determinant. In this paper, we study the determinants of locally positive semidefinite matrices and derive sharp lower bounds on their determinants that quantify the gap between local and global positive semidefiniteness. We further obtain analogous extensions of classical determinant inequalities, including Fisher and Koteljanskii inequalities, providing tight lower bounds in each case. In a sense, these results quantify, via determinant bounds, how far the class of locally positive semidefinite matrices can be from being positive semidefinite.

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

1 major / 0 minor

Summary. The manuscript studies (n-1)-locally positive semidefinite matrices (every proper principal submatrix is PSD) and claims to derive sharp lower bounds on their determinants that quantify the gap to global positive semidefiniteness. It further extends the classical Fisher and Koteljanskii determinant inequalities to this class, providing tight lower bounds in each case.

Significance. If the central claims hold under appropriate normalization, the results would provide new, sharp determinant inequalities that measure the relaxation gap in the local-to-global PSD hierarchy, with potential utility in semidefinite programming and combinatorial optimization. The extensions of Fisher and Koteljanskii inequalities would be a notable contribution if they are indeed tight and correctly derived.

major comments (1)
  1. [Abstract and main theorem statements] The (n-1)-local PSD property is invariant under positive scaling tA for t > 0, but det(tA) = t^n det(A). Consequently, if the class contains any matrix with negative determinant, det can be driven to -∞ by scaling, so no finite lower bound exists over the un-normalized class. The abstract and main claims (e.g., the statement of sharp lower bounds and the extensions of Fisher/Koteljanskii) therefore require an explicit normalization condition (unit diagonal, unit trace, or fixed (n-1)-minors) whose presence, homogeneity, and compatibility with sharpness must be verified; this appears unstated in the provided framing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the critical issue of normalization. We agree that the scaling invariance of the (n-1)-local PSD property combined with the homogeneity of the determinant requires an explicit normalization condition for the lower bounds to be well-defined and finite. We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The (n-1)-local PSD property is invariant under positive scaling tA for t > 0, but det(tA) = t^n det(A). Consequently, if the class contains any matrix with negative determinant, det can be driven to -∞ by scaling, so no finite lower bound exists over the un-normalized class. The abstract and main claims (e.g., the statement of sharp lower bounds and the extensions of Fisher/Koteljanskii) therefore require an explicit normalization condition (unit diagonal, unit trace, or fixed (n-1)-minors) whose presence, homogeneity, and compatibility with sharpness must be verified; this appears unstated in the provided framing.

    Authors: We fully agree with this observation. The (n-1)-local PSD property is indeed invariant under positive scaling, while the determinant scales by t^n, so without normalization the determinant can be made arbitrarily negative. Our derivations are performed under the normalization that all diagonal entries are equal to 1 (i.e., the matrices are correlation matrices in the local PSD sense). This choice is compatible with the classical Fisher and Koteljanskii inequalities and allows the extremal examples to achieve the stated sharp bounds. However, we acknowledge that the normalization was not explicitly declared in the abstract or the statements of the main theorems. In the revised version we will (i) add an explicit normalization sentence to the abstract, (ii) restate the main theorems with the unit-diagonal assumption clearly indicated, and (iii) confirm that all sharpness-attaining matrices satisfy this normalization and that the bounds remain homogeneous under the chosen scaling. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived directly from (n-1)-local PSD definition

full rationale

The paper derives sharp lower bounds on determinants (and extensions of Fisher/Koteljanskii inequalities) for matrices whose proper principal submatrices are PSD while the full matrix satisfies only a determinant condition. No quoted step reduces a claimed prediction or bound to a fitted parameter, self-defined quantity, or self-citation chain by construction. The derivation starts from the explicit locality assumption and produces new quantitative gaps to global PSD; the provided abstract and description contain no self-referential definitions or load-bearing prior results from the same authors. The skeptic scaling observation concerns whether a finite bound can exist without an unstated normalization (e.g., fixed diagonal or trace), but that is a question of claim validity rather than circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the standard definition of positive semidefiniteness and the algebraic properties of principal submatrices and determinants; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Every proper principal submatrix of an (n-1)-locally PSD matrix is positive semidefinite by definition.
    This is the defining property of the matrix class under study.
  • standard math Determinant is a continuous multilinear function of the matrix entries.
    Used implicitly when deriving lower bounds and extensions of classical inequalities.

pith-pipeline@v0.9.0 · 5427 in / 1354 out tokens · 32692 ms · 2026-05-12T01:17:23.116053+00:00 · methodology

discussion (0)

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