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REVIEW 3 major objections 5 minor 20 references

The maximum likelihood degree of the Grassmannian of lines Gr(2,n) is (2^{n-1} - n)(n-3)!.

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2026-08-01 12:18 UTC pith:Q3AXR4BR

load-bearing objection Friedman proves the closed-form ML degree of Gr(2,n) — a real new result with an elegant proof strategy — but one load-bearing algebraic step in Proposition 2.1 is skipped and needs to be written out. the 3 major comments →

arxiv 2607.19593 v2 pith:Q3AXR4BR submitted 2026-07-21 math.AG math.COmath.STstat.TH

Maximum Likelihood Estimation on the Grassmannian of Lines

Hannah Friedman This is my paper
classification math.AG math.COmath.STstat.TH
keywords grassmannianlineslikelihoodmaximumpositivealgebraicclosedcritical
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.

The paper studies a statistical model where the data are subsets of size 2 (pairs) of {1,...,n}, and the probability of a pair is proportional to the corresponding Plücker coordinate on the Grassmannian Gr(2,n). Given observed counts u, maximum likelihood estimation asks for the distribution in this model that makes the data most probable. For generic data, the likelihood equations have finitely many complex solutions; their number is the maximum likelihood degree (ML degree), a measure of how hard the optimization problem is.

The main theorem gives a closed formula for this degree: (2^{n-1} - n)(n-3)!. The proof translates the problem into topology. The ML degree equals the signed Euler characteristic of the Grassmannian minus a hypersurface. The author shows that the troublesome discriminant does not meet the open Grassmannian, so the count factorizes into a known quantity for the moduli space of n points on the projective line times a volume of the (2,n)-hypersimplex. A new combinatorial lemma about signed excedance enumerators makes the determinant computation work.

The paper also investigates real and positive solutions for Gr(2,4). It computes the logarithmic discriminant, which separates different real-solution count regimes, and finds a small region of the data simplex where the likelihood function has three positive critical points. This refutes an earlier conjecture of the author that positive data always gives at most one positive critical point. The numerical and symbolic computations are available online.

Core claim

Theorem 1.1: The Grassmannian Gr(2,n) has ML degree (2^{n-1} - n)(n-3)!. If the paper is correct, the number of complex critical points of the log-likelihood function for generic data on the positive Grassmannian of lines is given by this closed formula for every n.

Load-bearing premise

Lemma 2.2, especially the T = S+1 case whose determinant is quoted from [16, Theorem 6], is the combinatorial engine of Proposition 2.1. The monomial form of det(X) on Gr(2,n) -- and hence the constancy of the fiber Euler characteristic that yields Theorem 1.1 -- collapses if this enumerator identity is wrong or misapplied at q = -1. The paper states the identity but does not prove the S+1 block determinant, deferring to [16].

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

3 major / 5 minor

Summary. The paper presents a closed formula for the maximum likelihood degree of the Grassmannian of lines, Gr(2,n), claiming MLdegree = (2^{n-1} - n)(n-3)!. The proof computes the signed Euler characteristic of Gr(2,n)\H by fibering over the moduli space M_{0,n}. The key step is Proposition 2.1, which asserts that the principal A-determinant of the second hypersimplex restricts to a nonzero monomial on the open Grassmannian; this is derived from a determinant computation for a symmetric matrix whose entries are Plücker coordinates. The final section reports numerical experiments on real and positive critical points, including data indicating that the log-likelihood function can have three positive critical points in Gr(2,4).

Significance. If correct, Theorem 1.1 resolves a case of a known open problem and provides a simple closed formula that matches all previously known values (n=4:4, n=5:22, n=6:156). The proof strategy is elegant and brings together Euler characteristic methods, A-determinants, and known results for M_{0,n}. The manuscript is accompanied by reproducible software, and the numerical claims are concrete and falsifiable. The main theorem is likely correct, but the proof of Proposition 2.1 contains a load-bearing omission that must be addressed before the paper can be accepted.

major comments (3)
  1. [Section 2, Proposition 2.1] The transition from the Lemma 2.2 coefficient evaluation to the displayed formula for det(X) is not justified. After evaluating the restricted excedance enumerator at q=-1, the determinant is a sum over two monomial families: u_S v_S u_{S^c} v_{S^c} (with coefficient 2^{n-1} for S=∅,[n] and 2^{n-2} for proper nonempty S) and u_S v_{S+1} u_{S^c+1} v_{S^c} (with coefficient 2^{n-2}). The proof then jumps to an expression containing only the S+1 family, with no derivation of how the T=S contributions cancel or are absorbed. This is exactly the step that makes det(X) a monomial, and hence makes the fiber Euler characteristic constant in Theorem 1.1. The gap is proof-completeness, not necessarily a false result, but it must be filled with an explicit algebraic derivation or a direct proof of the displayed identity.
  2. [Section 2, Lemma 2.2] The case T=S+1 is the combinatorial engine of Proposition 2.1, but its determinant evaluation is deferred to [16, Theorem 6] with no further argument. The proof of Lemma 2.2 states 'see [16, Theorem 6]' without explaining how that theorem applies to the two possible block forms (when n∈S and when n∉S). Since Proposition 2.1 relies on this identity, the author should give a self-contained proof or at least a precise statement of the cited theorem and its application. This is not a blocking issue if the citation is correct, but the current level of detail is insufficient for a central step.
  3. [Section 3, Proposition 3.1] The proof asserts that a union of 35 hyperplanes that do not meet the relative interiors of any edges of the 5-simplex cannot intersect the open simplex. This geometric claim is true but not proved; a hyperplane cutting the interior of a simplex does have a vertex in the relative interior of some edge, but the fact should be stated or referenced. Without justification, the constancy of the number of positive critical points is not fully established. This does not affect Theorem 1.1 but is part of the Section 3 claims.
minor comments (5)
  1. [Section 2, Lemma 2.2] The statement overlaps for S=∅ and S=[n]: the second bullet '(1-q)^{n-2} if T=S' should specify ∅⊊S⊊[n] to avoid ambiguity.
  2. [Section 2, Eq. (5)] The coefficient computation for u_[n]v_[n] is correct, but the parity split is not derived. A one-line binomial sum would help the reader.
  3. [Section 3] The acknowledgment mentions counterexamples found by large language models. The actual counterexample data or a reference to a supplement should be included so the claim is independently verifiable.
  4. [Table 2] The row 'Boundary points' is cryptic. Define e_{ij} and explain how the regions are represented and sliced.
  5. [Section 2, Proposition 2.1] The notation '(-1)^{i<j}' is nonstandard and explained only in words. A sign function such as ε_{ij} = -1 if i<j and +1 if i>j would improve readability.

Circularity Check

0 steps flagged

No circular reduction: Theorem 1.1 rests on independent Euler-characteristic, volume, and excedance-determinant inputs; self-citations are contextual. A proof gap in Proposition 2.1 is a completeness risk, not circularity.

full rationale

The central derivation chain is not circular. Theorem 1.1 is obtained from [13, Thm 1.7] (ML degree = signed Euler characteristic), the quotient map to M_{0,n}, the fiber Euler characteristic via the hypersimplex volume, non-intersection of the principal A-determinant with the open Grassmannian, and the external enumerator identity in Lemma 2.2 quoted from [16]. None of these inputs asserts the Gr(2,n) ML degree formula; no fitted parameter is renamed as a prediction. The author's own [8] and [9] appear only for numerical context and comparison (Gr(3,6) and the squared Grassmannian table), not in the proof of Theorem 1.1, so they are not load-bearing self-citations. There is a genuine proof-completeness gap at Proposition 2.1: after Lemma 2.2 is evaluated at q = -1, the determinant still contains T=S contributions (coefficient 2^{n-1} for S=∅,[n] and 2^{n-2} for proper S), and the paper jumps directly to 'In either case, the determinant evaluates to det(X)=(-1)^n ... =(-2)^{n-2} ...' without deriving the cancellation of those T=S terms or the absorption of the extreme coefficients. This is a missing algebraic step, not a circular reduction: the identity is an independently checkable polynomial identity in u,v, and Lemma 2.2 is an external combinatorial determinant result. The gap affects verification of Theorem 1.1 but does not make the conclusion equivalent to its inputs. Overall, no circular derivation was found; the modest score reflects only the two non-load-bearing self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The central theorem has no fitted parameters and introduces no new entities. It rests on standard theorems in likelihood geometry, toric geometry, and combinatorics, plus the reliability of two established software packages for the secondary numerical results.

axioms (7)
  • standard math ML degree equals the signed Euler characteristic of the very affine variety Gr(d,n) minus H ([13, Theorem 1.7])
    Invoked at the start of Section 2 to turn the ML degree into a topological quantity.
  • standard math The Euler characteristic of the fiber F_x equals (-1)^{n-1} Vol(Delta_{2,n}) when the principal A-determinant does not vanish ([2, Theorem 2]; [10, Theorem 6.2.4])
    Justifies the constant fiber Euler characteristic used in the multiplicative computation.
  • standard math The principal A-determinant of Delta_{2,n} is the product of principal minors of size at least 4 of the matrix X ([6, Theorem 3.6])
    Reduces the A-determinant to the determinants evaluated in Proposition 2.1.
  • standard math chi(M_{0,n}) = (-1)^{n-3}(n-3)! ([17, Proposition 1])
    Supplies the base Euler characteristic for the fiber bundle computation.
  • standard math Restricted signed excedance enumerator identities of Lemma 2.2, with the T=S+1 case taken from [16, Theorem 6]
    Combinatorial input to Proposition 2.1; the S+1 determinant is quoted rather than proved.
  • standard math Multiplicativity of Euler characteristic for maps of complex algebraic varieties with constant fiber Euler characteristic ([7, Theorem 3.2.3])
    Used to compute chi(Gr(2,n) minus H) as the product of base and fiber Euler characteristics.
  • domain assumption Reliability of the numerical irreducible decomposition and symbolic computations in HomotopyContinuation.jl and Oscar.jl for the d=3,6 and Section 3 results
    Theorem 2.4 and Proposition 3.1 depend on numerical computations; the authors themselves note that nonreduced components may be missed.

pith-pipeline@v1.3.0-alltime-deepseek · 10372 in / 30507 out tokens · 205547 ms · 2026-08-01T12:18:38.003060+00:00 · methodology

0 comments
read the original abstract

We study the positive Grassmannian through the lens of algebraic statistics. A closed formula is presented for the maximum likelihood degree of the Grassmannian of lines. We study the real and positive critical points for the Grassmannian of lines in 3-space.

Figures

Figures reproduced from arXiv: 2607.19593 by Hannah Friedman.

Figure 1
Figure 1. Figure 1: Empirically observed real critical point counts of the log-likelihood function on [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The logarithmic discriminant (green) and Grassmannian Gr(2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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Reference graph

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