Closed Form of a Generalized Sinkhorn Limit

Max Chicky Fang

REVIEW 2 major objections 7 minor 15 references

Closed Form of a Generalized Sinkhorn Limit

T0 review · 2 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper derives an explicit formula for the generalized Sinkhorn limit of positive $2 \times 2$ matrices with arbitrary positive target sums, and proves that every entry of the generalized Sinkhorn limit of any positive $n \times m$…

desk verdict A genuinely new 2x2 closed form, but the claimed degree bound for all matrices rests on a dimension argument that fails for special positive inputs—worth reviewing, not accepting as-is. read the letter →

arxiv 2506.06338 v2 pith:UMM2YW5K submitted 2025-05-31 math.GM

Max Chicky Fang This is my paper

classification math.GM MSC 14N0513P1015A99

keywords generalizedSinkhornlimititerativematrixscalingalgebraicdegreeboundSegrevarietyconeGröbnerbasis2x2closedformprojectiveintersection

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

Iterative matrix scaling to prescribed row and column sums has no general closed form, and this paper pins down its algebraic nature. It derives an explicit formula for the generalized Sinkhorn limit of a positive nonsingular $2 \times 2$ matrix with arbitrary positive target sums, covering the singular case by a continuity argument. It then proves that for any positive $n \times m$ matrix with positive target row and column sums, every entry of the limit is algebraic over the field generated by the input data, with degree at most $\binom{n+m-2}{n-1}$. The point of the result is that a black-box iteration is replaced by exact algebraic structure: limit entries are roots of bounded-degree polynomials, and the bound matches a natural combinatorial count.

What carries the argument

The load-bearing object is the projective cone over the Segre variety, $\mathcal{C}(\Sigma_{n-1,m-1})$, inside $\mathbb{P}^{nm}$; the Segre variety is the image of pairs of projective vectors under the outer-product map. Scaling factors $r_i, c_j$ form a rank-one matrix $v_{ij}=r_i c_j$, and the cone encodes all such rank-one matrices up to scale together with a homogenizing coordinate. The row and column sum equations become $n+m-1$ independent linear equations defining a projective subspace $L$, so the solution set is the scheme-theoretic intersection $H=\mathcal{C}(\Sigma_{n-1,m-1}) \cap L$. The degree of this intersection, $\binom{n+m-2}{n-1}$, bounds the degree of the field extension generated by the entries of the limit. In the $2 \times 2$ case, the same system is solved explicitly with a Gröbner basis after a gauge fix, producing the square-root formula.

What would settle it

Use a computer algebra system on a $3 \times 3$ positive rational matrix with rational target sums: solve the Gröbner basis of the scaling equations and inspect the degree of the resulting univariate polynomial for any limit entry; a degree above $6$ would disprove the claimed bound. A more direct diagnostic is to exhibit an input, such as the all-ones $2 \times 2$ matrix the paper discusses, where the intersection $\mathcal{C}(\Sigma_{n-1,m-1}) \cap L$ has a positive-dimensional component, contradicting the dimension-zero premise.

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Extended reading notes

Core claim

The paper's central claim is that the generalized Sinkhorn limit—the limiting matrix obtained by repeatedly rescaling rows and columns to prescribed positive sums—has an explicit closed form in the smallest nontrivial case and a uniform algebraic bound in all cases. For a positive nonsingular $2 \times 2$ matrix, the limit entries are rational expressions in the four input entries and the target row and column sums together with one square root; the formula comes from fixing one scaling variable and computing a Gröbner basis for the four quadratic equations imposed by the row and column sums. When all target sums equal $1$, the formula reduces to the known doubly stochastic $2 \times 2$ limit, and by continuity it extends to singular $2 \times 2$ matrices, where the limit depends only on the target sums. The broader theorem states that for any positive $n \times m$ matrix and positive target sums, each entry of the limit is algebraic over the field $\mathbb{Q}(A, \vec{R}, \vec{C})$ with minimal polynomial degree at most $\binom{n+m-2}{n-1}$. The proof interprets the scaling factors as the rank-one outer product $r_i c_j$, homogenizes the system, and bounds the field-extension degree by the degree of the intersection of the projective cone over a Segre variety with a linear subspace.

Load-bearing premise

The degree-bound proof assumes that the row- and column-sum equations always cut the solution set down to finitely many isolated points; for special positive matrices the subspace can fail to sit in general position, leaving positive-dimensional components, and the paper leaves those cases unhandled.

Editorial extensions

If this is right

Every generalized Sinkhorn limit of a positive matrix with positive targets has entries in a finite extension of the input field, so no entry can be transcendental or require algebraic degree above the stated binomial bound.
The bound reproduces the small computed cases—degree $1$ for $1 \times n$, $2$ for $2 \times 2$, $3$ for $2 \times 3$, and $4$ for $2 \times 4$—consistent with the same combinatorial count.
The explicit $2 \times 2$ formula specializes to the classical doubly stochastic case when all target sums are $1$, and its limiting form covers singular matrices.
For singular $2 \times 2$ inputs the limit is independent of the input matrix and is determined entirely by the target sums, matching a maximum-entropy solution.
Fully simplified closed forms for larger matrices become impractical, so the algebraic-degree theorem, rather than the explicit formulas, is the broadly applicable statement.

Reading between the lines

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

A natural reading is that the degree bound holds on a Zariski-open set of inputs, with special-position inputs lowering the degree; the singular $2 \times 2$ formula, whose entries are rational, is one instance of that drop.
The same cone-intersection degree count should extend to tensor scaling, where the Segre cone degree becomes a multinomial coefficient and would bound the algebraic degree of tensor scaling limits.
A concrete test: for random rational $3 \times 3$ inputs, numerical limit entries should have minimal polynomials of degree at most $6$, with degree $6$ occurring generically.
Classifying the non-general-position cases could split the theorem into an upper bound plus a rationality criterion, explaining when iterative scaling produces rational output.

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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

Short version: this paper gives a genuinely new closed form for the 2x2 generalized Sinkhorn limit with arbitrary positive targets, and then claims a much bigger theorem about algebraic degree bounds for all n x m matrices. The 2x2 formula is the real content; the degree bound is not proved.

The 2x2 result is solid. It extends Nathanson's targets-1 formula to arbitrary positive row and column sums, and the algebra checks out: the expressions satisfy the marginal constraints and reduce to Nathanson's formula when targets are all 1. The derivation via Groebner basis is transparent, and the corollary for singular matrices via L'Hopital is a nice, valid extension. The author also provides Mathematica and Python code, which made it easy to verify the numerical claims. Credit where due: this is reproducible work.

The soft spots are real but uneven. The branch selection in the 2x2 formula (minus square root) is justified by a single numerical test. That is not a proof, though it is likely fixable with a sign analysis of the discriminant and the physical requirement r2 > 0. Minor.

The bigger problem is Theorem 3.1.1. The proof asserts the scheme H = C(Segre) ∩ L has dimension 0 by a dimension count. That count assumes general position. For special positive inputs, L is not in general position: the stress-test example with a 2x3 all-ones matrix and targets R=(1,1), C=(2/3,2/3,2/3) gives a line at infinity inside the intersection, so dim H >= 1. The author explicitly concedes in Section 3.2 that this case is 'likely' and unhandled, and the parenthetical claim that 'accounting multiplicity returns the same result' does not hold when a positive-dimensional component is present. So the degree bound remains a conjecture, not a theorem.

The paper is honest about its limitations—arguably too informal in places—but the central proof gap is load-bearing. If the dimension issue can be fixed under a genericity assumption, or if the theorem is restated for 'generic' A, R, C, the degree bound might go through. As written, it doesn't.

Who is this for? Researchers interested in the algebraic structure of matrix scaling. The 2x2 formula is a worthwhile small contribution. The degree bound is important but still open.

Recommendation: send this to a serious referee. The small result deserves review, and the large claim deserves scrutiny of exactly the kind the stress-test provides. I'd ask for major revision: either prove the dimension statement with a proper genericity argument, or state the theorem as a conditional result and make the 2x2 proof rigorous on the branch.

Referee Report

2 major / 7 minor

Summary. The paper studies the generalized Sinkhorn (Kruithof) limit S = D1 A D2 of a positive n x m matrix A with prescribed positive row and column sums R and C. It gives explicit closed forms for the 1 x n case and for nonsingular 2 x 2 matrices, together with a singular-limit corollary obtained by L'Hôpital, and it claims a general theorem that every entry of the generalized Sinkhorn limit is algebraic over Q(A,R,C) with degree at most binomial(n+m-2,n-1). The proof of the general theorem interprets the homogenized scaling equations as an intersection H of the projective cone C(Σ_{n-1,m-1}) over a Segre variety with a linear subspace L in P^{nm}, and uses the degree of that intersection to bound the field extension degree.

Significance. The 2 x 2 closed form, if fully proved, would be a useful explicit generalization of Nathanson's doubly stochastic formula to unequal targets, and the algebraic-degree theorem would establish the Rowland-Wu conjecture in the generalized setting. The paper is commendably transparent about its computational workflow and includes reproducible Mathematica and Python code. However, the central theorem is not proved as written: the intersection H need not be zero-dimensional for arbitrary positive data, and the branch choice in the 2 x 2 formula is justified only by a single numerical test. The 2 x 2 result appears salvageable with elementary algebra, and the degree bound may be true, but a substantially repaired argument is required.

major comments (2)

[Section 3.1, proof of Theorem 3.1.1] The assertion "This scheme H has dimension 0 as expected" is false for arbitrary positive inputs. For n=2, m=3, A=ones(2,3), R=(1,1), C=(2/3,2/3,2/3), the homogenized equations are v11+v12+v13=t, v21+v22+v23=t, and v11+v21=v12+v22=v13+v23=2t/3. In the hyperplane t=0 these reduce to v21=-v11, v22=-v12, v23=-v13 and v11+v12+v13=0; every rank-1 matrix of the form [[x,y,-x-y],[-x,-y,x+y]] lies in C(Σ_{1,2})∩L, so H contains a positive-dimensional family of points. Hence dim H≥1, and the degree bound [K(V):K]≤binom(n+m-2,n-1), which relies on dim H=0, does not follow. The remark that "when the subspace is not in a general position, accounting multiplicity returns the same result" is not valid for a non-proper intersection with positive-dimensional components; Section 3.2 itself concedes that this case is likely and unhandled.
[Section 2.3, Eqs. (2.3.3) and Theorem 2.3.1] The proof selects the minus square root in Eq. (2.3.3) because, for the single numerical test A={{1,2},{3,4}} and R=C=(1,1), the plus root is negative (Appendix 4.1.5). A proof of Theorem 2.3.1 needs to show for every positive nonsingular 2x2 matrix A and every positive R,C that the discriminant is nonnegative and that the minus branch gives the unique positive solution, equivalently that the plus branch is extraneous. The proof should also verify directly that the displayed matrix has row sums R and column sums C; the current proof merely refers to the Groebner basis computation and says "Prior in this section." Without this sign and positivity analysis, the closed form is not established as a theorem for all inputs covered by the statement.

minor comments (7)

[Section 2.2, proof of Theorem 2.2.1] The proof "Just jiggle it a bit and the proof should pop out" is not an acceptable proof; this one-line derivation should be written out explicitly.
[Section 2.3, displayed Groebner basis] The displayed Groebner basis contains garbled terms such as "a11a21c2 1C2" and "a11a2 22r2"; the correct expressions appear in Appendix 4.1.1, and the display should be corrected.
[Section 3.1, notation and dimension computation] The expression "C(Σ_{n-1,m-1}) ∈ P^{nm}" should read "C(Σ_{n-1,m-1}) ⊂ P^{nm}", and the dimension computation should explicitly state which intersections are assumed to be proper and generic versus which hold for all inputs.
[Section 3.2] Informal remarks such as "Also I'm... just not gonna bother with 2 × 3 its way too messy bruh" and "my brain is absolutely too cooked" are inappropriate in a formal paper and should be removed.
[Section 3.2] The discussion of Bring radicals, hypergeometric series, and possible closed forms for septic and octic equations is speculative and unrelated to the proved results; it should be cut or explicitly labeled as speculation.
[Corollary 2.3.1] The phrase "WLOG choose a12 to be a continuous function" is imprecise; it should say "choose a one-parameter family A(ε) of positive nonsingular matrices converging to A0."
[References] Reference [9] is a private Discord conversation; it should be replaced by a citable source or removed, since it cannot be verified by readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 2x2 closed form is derived from the defining equations and the degree theorem is proved with external algebraic geometry; the non-general-position gap is a soundness issue, not circularity.

full rationale

The paper's derivations are self-contained rather than circular. Theorem 2.3.1 is obtained by writing down the defining equations (2.1.3) for the generalized Sinkhorn limit, computing a Groebner basis, and solving by back-substitution; the formula is a solution of those equations rather than an input assumed into them. The choice of the negative square-root branch is tested on one matrix, which is an unsupported empirical step, but it is not a fitted parameter masquerading as a prediction. Theorem 3.1.1 uses standard external algebraic-geometry facts — the degree of the Segre variety and the degree bound for zero-dimensional schemes — and targets the independently stated Rowland-Wu conjecture; none of these reduce to the paper's own conclusion. There is no load-bearing self-citation: the single author cites Rowland and Wu and Sinkhorn externally, and the private conversation [9] is not used as evidence. The real weakness is a soundness gap, not circularity: Section 3.1 asserts that H = C(Sigma_{n-1,m-1}) ∩ L has dimension 0 'as expected' and that non-general-position subspaces can be handled by multiplicity, while Section 3.2 explicitly concedes that arbitrary positive A, R, C probably put L in non-general position (for a positive rank-1 2x3 matrix, H contains a line at infinity), so the degree bound is not proved for all positive inputs. Thus Theorem 3.1.1 is not established for the full stated domain, but its failure mode is an unproved genericity assumption, not an equation reducing to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard algebraic geometry facts, on the existence/continuity of Sinkhorn limits from prior literature, and on one ad hoc assumption about the dimension of the intersection H that is not proven and is false in some cases.

assumptions (4)

standard math Degree of the Segre variety Σ_{n-1,m-1} is C(n+m-2, n-1)
Cited from Harris [2] and MathOverflow [15] in the proof of Theorem 3.1.1.
standard math For a zero-dimensional projective scheme of degree d, the residue field of each closed point has degree at most d
Cited from MathStackExchange [6]; standard in intersection theory.
domain assumption The generalized Sinkhorn limit exists, is unique, and is continuous in the input matrix
Sinkhorn [13] and Franklin-Lorenz [1], used in Corollary 2.3.1 and throughout.
ad hoc to paper The intersection H = C(Σ_{n-1,m-1}) ∩ L is zero-dimensional for all positive A,R,C
This is asserted in the proof of Theorem 3.1.1 without proof; it is false in some cases (all-ones matrix gives a positive-dimensional component at infinity), so the proof is incomplete.

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Cite this review

Pith. "Pith review of Closed Form of a Generalized Sinkhorn Limit." pith.science (2026). https://pith.science/paper/UMM2YW5K

@misc{pith2026250606338,
  author       = {Pith},
  title        = {Pith review of: Closed Form of a Generalized Sinkhorn Limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMM2YW5K}},
  note         = {Machine review of arXiv:2506.06338}
}

abstract

The Kruithof iterative scaling process, which adjusts matrices to meet target row and column sums, is a longstanding problem that lacks a general closed form for its limit. While Nathanson derived the closed form for the Sinkhorn limit of $2\times 2$ matrices when target row and column sums are 1, and recent work by Rowland and Wu has advanced understanding of Sinkhorn limits for $3\times 3$, and general $n\times m$ matrices through polynomials, a "generalized Sinkhorn limit" (i.e. the original "Kruithof limit", with arbitrary target sums) remains elusive. Here, we derive the closed form for the generalized Sinkhorn limit of $2\times 2$ matrices, and discuss how this approach can be extended to larger matrices. More significantly, we prove that for any positive $n \times m$ matrix and positive target row and column sums, each entry in the generalized Sinkhorn limit is algebraic over the input data with degree at most $\binom{n+m-2}{n-1}$.

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Reference graph

Works this paper leans on

15 extracted references · 14 canonical work pages

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