Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial

Leandro Farina; Tiago Closs

REVIEW 3 major objections 5 minor 17 references

Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper claims that the three highest-order characteristic-polynomial coefficients (a_{n-1}, a_{n-2}, a_{n-3}), which are signed sums of all 1x1, 2x2, and 3x3 principal minors, already separate totally positive from non-totally positive m

desk verdict Clear empirical finding for the sampled families, but the TP/non-TP claim outruns the data because the negative class is too easy; worth refereeing with hard-negative tests. read the letter →

arxiv 2607.18148 v1 pith:N3GPUANL submitted 2026-07-20 cs.LG math.RA

Tiago Closs , Leandro Farina This is my paper · ORCID

classification cs.LGmath.RA MSC 15B4815A1862H3068T07

keywords totalpositivitycharacteristicpolynomialcoefficientsprincipalminorsMahalanobisellipsoidsstructuredmatricesfeatureattributionspectralinvariantsneuralnetworkclassifiers

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

This paper tries to establish that total positivity of a matrix — the requirement that every minor be positive — can be read off, for practical purposes, from just three numbers: the three highest coefficients of the characteristic polynomial. Because those coefficients are, up to sign, the sums of all 1x1, 2x2, and 3x3 principal minors, they aggregate exactly the low-order minor information that total positivity constrains. In experiments on matrices of size 5, 10, and 30, classifiers given only these three numbers separated totally positive from non-totally positive matrices with accuracy above 0.998, and each structured family (bidiagonal products, Vandermonde, Cauchy) occupied its own ellipsoidal cloud in this three-dimensional space, with the clouds becoming disjoint as the dimension grows. If true, this would give a tractable geometric screening test for a property whose direct verification requires checking an exponential number of minors.

What carries the argument

The engine of the argument is the projection π(A)=(a_{n-1},a_{n-2},a_{n-3}) combined with the identity a_{n-k}=(-1)^k E_k(A), where E_k is the sum of all k×k principal minors. This links the three chosen scalars directly to the minor conditions that define total positivity. On top of that projection, the paper fits Mahalanobis ellipsoids — sets {x:(x−μ)^T Σ^{-1}(x−μ) ≤ τ} — to each family's point cloud, using the mean and covariance of the coefficients to give the cloud a concrete geometric shape. The three TP families exploited are products of oscillatory bidiagonal pairs (each pair is an oscillatory matrix, and n pairs guarantee total positivity), Vandermonde matrices with sorted positive

What would settle it

Check all minors exactly for any positive-entry matrix that the three-coefficient classifier labels TP: a single non-positive minor would be a concrete counterexample to the claimed separation. Symmetrically, sample totally positive matrices outside the three generative families (for instance Pascal matrices or general oscillatory powers) and test whether their (a_{n-1},a_{n-2},a_{n-3}) projections land inside the fitted ellipsoids; a TP matrix landing far outside would show the signatures are family-specific rather than intrinsic to total positivity.

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Extended reading notes

Core claim

The paper's central claim is that the map π(A)=(a_{n-1},a_{n-2},a_{n-3}) — the three highest-order characteristic-polynomial coefficients — is a concise spectral fingerprint of total positivity. Up to sign, these coefficients are the elementary symmetric sums E_1, E_2, E_3 of the principal minors of orders 1, 2, and 3. Using feedforward networks trained on only these three inputs, the authors obtain test accuracies of 0.9998 (n=5), 0.9998 (n=10), and 0.9999 (n=30) on the binary TP/non-TP problem, while a linear SVM plateaus around 0.84 and an RBF-kernel SVM reaches about 0.99, showing the boundary is nonlinear but geometrically simple. In the same coordinate space, the TP samples from each o

Load-bearing premise

The results stand or fall with the assumption that the randomly sampled positive-entry non-TP matrices and the three generative TP families (with the n=30 bidiagonal guarantee resting on extensive sampling rather than exhaustive minor checks) are representative of the full classes of totally positive and non-totally positive matrices.

Editorial extensions

If this is right

Across n=5, 10, and 30, a classifier restricted to three coefficients reaches 0.9998–0.9999 test accuracy, so a three-number screen can flag total positivity with few errors for these structured families, bypassing the roughly 5×10^7 minors needed at n=30.
The failure of linear separation (0.805–0.877) and success of quadratic/RBF separation (~0.99) implies the TP/non-TP boundary in coefficient space is nonlinear but low-complexity.
The pairwise family overlap of the 95% Mahalanobis ellipsoids drops to 0% already at n=10 and stays 0% through n=100, while centroid Mahalanobis separation grows to tens of standard deviations for two of the three pairs.
Conjecture 1: with n large enough, the Mahalanobis ellipsoids enclosing any fixed fraction of each family's samples are pairwise disjoint, with separation bounded below in spread units.

Reading between the lines

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

If the separation is a general phenomenon, total positivity testing for large structured matrices could be reduced to evaluating three symmetric functions and checking one quadratic inequality, a dramatic cost cut relative to minor enumeration.
The paper's mapping suggests a testable extension: other structured TP classes (Toeplitz, Hankel, Pascal, or TP matrices built from other bidiagonal distributions) should also form concentrated ellipsoidal clusters; if they do not, the signatures are artifacts of the three generative recipes.
The observed quadratic boundary hints that the projection of the TP cone may have a semi-algebraic description of low degree; proving or refuting that would connect the numerics to algebraic geometry.
Because E_k = tr C_k(A) with C_k(A) the k-th compound matrix, the three-coefficient projection is equivalently a three-trace fingerprint of the first three compound matrices, suggesting other trace invariants (power sums, compound traces) as alternative or complementary fingerprints.

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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

Here's my take on arXiv:2607.18148. The paper makes a real empirical observation: for the three structured TP families it generates, the three highest characteristic-polynomial coefficients almost perfectly separate TP from random positive matrices. That part is solid; the numerical work is internally consistent, and the convergence study on the ellipsoid parameters is a reasonable stability check. The algebraic identity a_{n-k}=(-1)^k E_k is classical, and the paper uses it sensibly to motivate the feature choice. The ML/XAI part is handled honestly: it is exploratory, and the paper does not oversell it. The conjecture is clearly stated and the suggested proof route (compute limiting centroids and concentrate fluctuations) is plausible.

The soft spot is the one the stress-test flags, and it is load-bearing. The non-TP class is only random positive-entry matrices that fail total positivity. That excludes the natural near-boundary negatives, especially products of n-1 bidiagonal pairs, which are totally nonnegative but not totally positive and arise from the same generative scheme. The paper even states that total positivity first appears at exactly k=n-1 pairs, so these are the obvious adversarial examples, yet they never appear in the non-TP dataset. With only easy negatives, the 0.9998-0.9999 accuracy may be separating structured TP families from generic positive matrices, not total positivity. The abstract's wording overstates the finding.

There are secondary issues of the same kind. The n=30 TP guarantee is a structural theorem plus 'extensive sampling'; that is fine, but the non-TP labeling procedure at n=30 is not described, and exhaustive minor checking is impossible there. The Mahalanobis ellipsoids in Figure 9 and Table 2 are fit to the same samples they are said to enclose, and the rule for tau is not stated; the family-overlap table also evaluates fitted ellipsoids on the same 5000 matrices used to fit them, so that part is descriptive rather than predictive. Code and data are not public.

None of this kills the paper's core empirical observation for the distributions actually sampled. But the central claim needs re-scoping or a serious hard-negative test. I'd send it to a referee who knows total positivity and ask for exactly those experiments, plus held-out ellipsoid evaluation and public code. The paper deserves referee time; it just needs to earn its title.

Referee Report

3 major / 5 minor

Summary. The paper investigates whether the three highest-order characteristic-polynomial coefficients (a_{n-1}, a_{n-2}, a_{n-3}) can separate totally positive (TP) matrices from non-totally-positive matrices. TP examples are generated from three structured families—products of positive bidiagonal matrices, Vandermonde matrices, and Cauchy matrices—while non-TP examples are random strictly-positive-entry matrices that fail total positivity. The authors train neural-network classifiers and SVMs on the three coefficients and report test accuracies of 0.9998–0.9999 for dimensions n=5, 10, and 30 (Section 5). They fit Mahalanobis ellipsoids to the TP clouds, observe that the TP/non-TP boundary is nonlinear (Section 6), and find family-specific ellipsoidal signatures whose pairwise overlap vanishes for n≥10 (Section 7.3). Based on this evidence, they formulate Conjecture 1 on the asymptotic pairwise disjointness of the three family ellipsoids.

Significance. The paper proposes a concise spectral fingerprint for structured families of totally positive matrices and a concrete, falsifiable conjecture about their geometric separation. The empirical protocol is largely transparent: dataset construction is described in detail, and the authors include a convergence analysis of the ellipsoid parameters (Section 7.2) and give some exact-arithmetic verification for small n. The central TP-versus-non-TP claim, however, is only as strong as the sampling of the 'non-TP' class. Because that class consists only of generic positive-entry matrices that fail total positivity, the reported near-perfect accuracy may reflect distribution separation rather than a property of total positivity itself. The family-separation conjecture is better supported since it is explicitly scoped to the three generative models, but its evaluation contains an in-sample component. The paper is potentially interesting and the conjecture is a genuine contribution, but the headline claim needs additional hard-negative testing before it can be accepted at face value.

major comments (3)

[Section 2 (Non-TP sampling)] The non-TP class is constructed only as random matrices with strictly positive entries that fail total positivity. This omits near-boundary totally nonnegative matrices, such as products of k=n−1 bidiagonal pairs, which the paper itself notes are TN but not TP (Section 2.1). With only generic random matrices as negatives, the classifier may separate structured TP families from unstructured random matrices rather than detecting the total-positivity condition. The abstract's claim about 'separating totally positive from non-totally positive matrices' is therefore overbroad. Please add hard-negative families (e.g., products of n−1 bidiagonal pairs, matrices with a single small negative minor) and report per-family accuracy; without this, the 0.9998–0.9999 test accuracies do not establish the stated claim.
[Section 2.1 (TP guarantee for n=30)] The assertion that products of k=n bidiagonal pairs are totally positive is verified exhaustively only for n≤8; for n=10 and n=30 the support is 'extensive sampling' rather than exact minor checks. Since the paper's headline accuracy includes n=30, a mislabeled training point could in principle alter the reported numbers. Please provide a proof of the structural guarantee (or a reference to the oscillatory-matrix result) and state how many n=30 samples were checked in exact arithmetic. This is load-bearing for the n=30 classification results.
[Section 7.3 and Table 3 (In-sample ellipsoid overlap)] The overlap percentages in Table 3 are computed as the worst-case fraction of points of one family inside the other family's 95% Mahalanobis ellipsoid, using the same 5000 matrices per family used to fit those ellipsoids. This is an in-sample evaluation: it describes the fitted object rather than demonstrating predictive separation. To support Conjecture 1, evaluate the overlap on held-out samples or via cross-validation. The convergence analysis in Section 7.2 addresses parameter stability but does not remove this in-sample bias.

minor comments (5)

[Section 7.1] Typo: 'charactheristic' should be 'characteristic'.
[Section 7.3] The verification of ellipsoid disjointness via a 50^3 grid over a bounding box is not a rigorous intersection test. A computational-geometry or semidefinite-programming based check would be more convincing, though not necessary for the conjecture's plausibility.
[Section 3.2] The statement that test accuracies were 'identical ... achieving a perfect accuracy of 1.0' would be more informative with standard errors or results over multiple seeds, given the stochastic training procedure.
[Data availability] The datasets and code are 'available from the corresponding author upon reasonable request.' For reproducibility, a public repository would be preferable, especially since the paper's conclusions are empirical.
[Section 6.2] The text says the RBF kernel reaches accuracies 'near 0.99,' but Table 1 lists 0.9945, 0.9962, and 0.9885. Consider reporting these values explicitly to avoid ambiguity.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity: held-out classification is genuine; the only circular phrasing is the descriptive TP-enclosing ellipsoid, which is a fit restated.

self definitional [Section 7.1 (Mahalanobis Distance and Ellipsoids), Figure 9]
"As shown in Figure 9, the ellipsoids capture all TP matrices, while most non-TP matrices lie outside the boundaries."

Immediately before, the ellipsoid is fitted to the TP samples: 'Given a set of such vectors corresponding to TP samples, their mean and covariance matrix μ∈R^3, Σ∈R^{3×3} are calculated' and τ is the radius of the fitted Mahalanobis ellipsoid. Therefore 'capture all TP matrices' is true by construction when τ is the maximum sample Mahalanobis distance (or a quantile chosen on the same sample); it restates the fit rather than testing it. The non-TP exclusion is empirical, and the paper's central classification result uses an independent test split, so this tautology is descriptive and not load-bearing.

full rationale

The paper's headline claim that (a_{n-1}, a_{n-2}, a_{n-3}) separate TP from non-TP matrices is supported by a genuine held-out evaluation: Section 5 reports FFNs trained on the three coefficients and tested on a 10^4-matrix test set, with accuracies 0.9998–0.9999. That accuracy is not a fitted parameter renamed as a prediction. The Mahalanobis-ellipsoid geometry in Section 7 is explicitly a fitted description of the TP point clouds; the statement that the fitted ellipsoid 'captures all TP matrices' is tautological, but the paper does not use it as a predictive result, and Conjecture 1 is explicitly conjectural. The main validity concerns are not circularity: the non-TP class is sampled only as generic positive matrices (Section 2) and omits near-boundary totally nonnegative matrices (k=n−1 bidiagonal products), and the n=30 TP guarantee rests on 'extensive sampling' rather than exhaustive minor checks; these affect representativeness and label confidence, not the logic of the derivation. The self-citation [8] appears only in a general list of related work and is not load-bearing. No central result reduces by construction to its own inputs.

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

The paper's conclusions are entirely empirical. The labels rest on generative models whose parameters and distribution choices are only partially specified, the ellipsoid-based separation is fit to the same samples it describes, and the TP guarantee for the largest dimension rests on sampling rather than exhaustive verification.

free parameters (4)

bidiagonal entry distribution
Section 2.1 says 'random positive entries' on the sub/superdiagonals without giving a distribution; the resulting coefficient cloud depends on this choice.
Vandermonde/Cauchy node cutoff ε = unspecified (U(ε,1))
Sections 2.2–2.3 sample parameters from U(ε,1) with ε>0 'a small constant'; the value is not reported and changes the spread of the resulting matrices.
number of bidiagonal pairs k = k = n
Section 2.1 sets k=n, one above the stated minimum n−1; this modeling choice affects diversity and the distribution of the coefficients.
Mahalanobis radius τ = 122.426 / 251.705 / 138.201 for n=30 (Table 2)
Section 7.1 introduces τ to define the ellipsoid, but the fitting rule is never stated; Table 2 reports fitted τ values, while the overlap analysis switches to τ=χ²_{3,0.95}. This is a fitted quantity central to the geometric claims.

assumptions (4)

domain assumption Products of at least n−1 oscillatory matrices are totally positive, and each L_i U_i pair is oscillatory.
Section 2.1 relies on this to label bidiagonal products as TP without checking all minors. The paper says it was verified exactly for n≤8 and by extensive sampling up to n=30, but no proof or exhaustive check is given for n=30.
domain assumption The generative distributions in Section 2 are representative of TP and non-TP matrices.
Non-TP data consists only of random positive-entry matrices that fail total positivity, and TP data only of three structured families; no coverage argument is provided, so the classification and conjecture describe these distributions unless the assumption holds.
standard math Standard total-positivity facts: all minors of TP matrices are positive; a_{n-k}=(-1)^k E_k; Vandermonde and Cauchy matrices with positive ordered parameters are TP.
These are classical results cited from Pinkus, Fallat–Johnson, and Karlin, used throughout the paper.
domain assumption Floating-point computation of characteristic-polynomial coefficients is accurate enough at n=30 for the reported separation.
Section 2.1 notes that high-order minors fall below machine precision after normalization; the characteristic coefficients inherit similar numerical risk, but the paper does not quantify it.

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Cite this review

Pith. "Pith review of Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial." pith.science (2026). https://pith.science/paper/N3GPUANL

@misc{pith2026260718148,
  author       = {Pith},
  title        = {Pith review of: Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N3GPUANL}},
  note         = {Machine review of arXiv:2607.18148}
}

read the original abstract

We investigate the extent to which totally positive matrices can be distinguished through the highest-order coefficients of their characteristic polynomials. To identify the most informative coefficients, we also employed neural-network classifiers together with feature-attribution methods. Using datasets built from several structured totally positive families, including products of positive bidiagonal matrices, Vandermonde matrices, and Cauchy matrices, we find that the coefficients (a_{n-1}, a_{n-2}, a_{n-3}) already contain strong discriminatory information for separating totally positive from non-totally positive matrices in dimensions 5, 10, and 30. The resulting separation is markedly nonlinear and admits a natural geometric description in the corresponding three-dimensional coefficient space by means of Mahalanobis ellipsoids. These ellipsoids enclose the totally positive samples while excluding most non-totally positive ones. Moreover, different structured totally positive families exhibit distinct ellipsoidal signatures, and the separation between these signatures increases with the dimension. These observations lead us to formulate a conjecture on the geometric separation of structured totally positive families in the space determined by the three highest-order characteristic coefficients.

Figures

Figures reproduced from arXiv: 2607.18148 by the authors.

**Figure 2.** Training and validation accuracy and loss curves for the FFN trained on derived features [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Integrated Gradients attributions for three matrix sizes: (a) [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Integrated Gradients attributions using only characteristic polynomial coefficients for (a) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Integrated Gradients attributions using the three highest–order characteristic polynomial [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: SHAP feature importance across matrix sizes. (a) 5 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: SHAP values using only characteristic polynomial coefficients across matrix sizes: (a) [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Separating Hyperplane in the Feature Space for SVM Classification of 5x5, 10x10 and [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: Mahalanobis ellipsoids for TP matrices of sizes 5 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: Convergence of the Mahalanobis ellipsoid parameters across different matrix dimensions. [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: Inter-family separation as a function of the dimension [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: Mahalanobis ellipsoids for TP matrices from the Bidiagonal, Vandermonde, and Cauchy [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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