Algebraic Constraints for Linear Acyclic Causal Models

Cole Gigliotti; Elina Robeva

REVIEW 1 major objections 5 minor 19 references

Algebraic Constraints for Linear Acyclic Causal Models

T0 review · 1 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A random vector comes from a LiNGAM on a DAG exactly when certain moment matrices have rank equal to the number of parents of each vertex.

desk verdict The main theorem is correct and closes an open characterization: for any DAG, the LiNGAM second- and third-order moment model is cut out by explicit rank conditions on the matrices M_v. read the letter →

arxiv 2505.00215 v2 pith:6CUABHZS submitted 2025-04-30 math.ST stat.TH

Cole Gigliotti , Elina Robeva This is my paper · ORCID

classification math.STstat.TH MSC 62R01

keywords LiNGAMcausaldiscoverythird-ordermomentsalgebraicstatisticsdeterminantalconstraintsstructuralequationmodelsdirectedacyclicgraphmethodof

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 proves a complete algebraic characterization of the observable second- and third-order moments of a linear non-Gaussian acyclic model (LiNGAM). For any directed acyclic graph $G$, a positive definite covariance matrix $S$ and a symmetric third-order moment tensor $T$ are realizable under $G$ if and only if, for every vertex $v$, a matrix built from the moments of $v$, its parents, and its non-descendants has rank exactly equal to the number of parents of $v$. The characterization reduces membership testing to explicit determinantal conditions, extends the previously known polytree case to arbitrary DAGs, and contains the Gaussian local Markov property as the special case where third moments vanish. This matters because it turns the causal question of whether data could have come from a given graph into a check of simple rank constraints, the same algebraic structure that already powers non-Gaussian causal discovery algorithms.

What carries the argument

The central object is the blocked moment matrix $M_v$, whose top rows are indexed by the parents of $v$, whose bottom row is indexed by $v$, and whose columns are indexed by non-descendants of $v$ for the covariance entries together with pairs $(\text{non-descendant}, \text{any vertex})$ for the flattened third-moment entries. The condition that $M_v$ has rank $|\mathrm{pa}(v)|$ is equivalent to the bottom row lying in the span of the parent rows, and those span coefficients become the structural coefficients $\lambda_{uv}$ of the LiNGAM. The same matrix simultaneously encodes the second-moment relations, namely the classical local Markov property recovered from the covariance block alone, and the third-moment relations that carry the additional directional information needed to identify the full DAG rather than its Markov equivalence class.

What would settle it

The theorem would be refuted by a triple $(G,S,T)$ with $S$ positive definite and $T$ symmetric where every matrix $M_v$ has rank $|\mathrm{pa}(v)|$ but $(S,T)$ is not in the image of the parametrization $S=(I-\Lambda)^{-T}\Omega^{(2)}(I-\Lambda)^{-1}$, $T=\Omega^{(3)}\bullet (I-\Lambda)^{-1}\bullet (I-\Lambda)^{-1}\bullet (I-\Lambda)^{-1}$ for any $\Lambda\in\mathbb{R}^E$ and diagonal $\Omega^{(2)},\Omega^{(3)}$. For the three-vertex DAG $1\to 2$, $1\to 3$, $2\to 3$, one can test this directly with a computer algebra system: the theorem predicts the model is cut out by the 2-minors of $M_2$ and 3-minors of $M_3$ saturated by the principal minors of $S$, so any point in that variety outside the parametrization image would disprove the sufficiency direction.

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Extended reading notes

Core claim

Formally, Theorem 3.1 states that $(S,T)$ belongs to the second- and third-order moment model $M_{\le 3}(G)$ if and only if each matrix $$M_v = \begin{pmatrix} S_{\mathrm{pa}(v),\mathrm{nd}(v)} & T_{\mathrm{pa}(v),\mathrm{nd}(v)\times V} \\ S_{v,\mathrm{nd}(v)} & T_{v,\mathrm{nd}(v)\times V} \end{pmatrix}$$ has rank $|\mathrm{pa}(v)|$. The necessity direction comes from the structural equation $X_v = \sum_{u\in\mathrm{pa}(v)} \lambda_{uv} X_u + \varepsilon_v$: because $\varepsilon_v$ is independent of every non-descendant $X_w$, the vector $(-1,\lambda_{\mathrm{pa}(v),v})$ is a left null vector of $M_v$, and positive definiteness of $S$ prevents the rank from dropping below $|\mathrm{pa}(v)|$. The sufficiency direction reads the regression coefficients $\lambda_{uv}$ off the linear dependence of the bottom row of $M_v$ on the parent rows, and then shows that the same coefficients diagonalize both the transformed covariance matrix and the transformed third-order moment tensor, recovering the independent error terms. The model is therefore cut out inside the positive definite cone by the vanishing of all $(|\mathrm{pa}(v)|+1)$-minors, giving a finite list of polynomial constraints that separate one DAG from another.

Load-bearing premise

The characterization stands or falls on the LiNGAM assumption that the noise variables are mutually independent with mean zero and finite third moments and that every common cause is included in the graph; if errors are dependent or a hidden confounder exists, the moment equations that force the matrices $M_v$ to drop rank no longer hold.

Editorial extensions

If this is right

Membership of a moment pair $(S,T)$ in the LiNGAM model for any DAG is decided by finitely many rank conditions, so model selection between candidate DAGs can be framed as checking which graph's matrices $M_v$ drop to the required ranks.
The covariance-only part of the matrices recovers the Gaussian local Markov property, so the theorem is a genuine extension of classical covariance constraints rather than a separate theory.
The polytree characterization from earlier work follows as a special case, and the new proof shows the polytree $2\times 2$ determinants are submatrices of the more general rank conditions.
Source and sink nodes can be detected by the same machinery: the rank-one matrices used in existing source-finding algorithms are a direct corollary, and a new sink-detection procedure based on the condition numbers of $M_v$ works in simulations.
Because the constraints are all minors of explicit matrices, they give concrete polynomial equations usable in goodness-of-fit tests for whether data follow any linear non-Gaussian acyclic model.

Reading between the lines

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

One natural next step the paper does not take is to turn the rank conditions into a model-selection score: among graphs that explain the covariance equally well, the graph whose $M_v$ ranks are closest to $|\mathrm{pa}(v)|$ would be preferred, and finite-sample thresholds could be derived from the singular-value behavior the authors illustrate.
The authors' remark that higher-order cumulants should behave analogously suggests a concrete testable extension: replacing third moments by fourth cumulants should yield an analogous rank characterization for models with symmetric errors, where third moments vanish and the current theorem gives no grip.
The discussion of latent-variable LiNGAM hints that eliminating third-moment blocks from these rank conditions, rather than adding them, might produce new constraints for hidden-variable models, including some that cannot be expressed by covariance determinants alone.
The ROC-style threshold experiment in the paper suggests the rank tests have practical statistical behavior worth formalizing, but no asymptotic distribution of the condition numbers is proven here; that would be needed for principled threshold selection.

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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

Have you seen arXiv:2505.00215? The headline: the main theorem is correct and it is a genuine completion of an open characterization. For any DAG G, M≤3(G) is cut out by rank conditions on the matrices M_v built from S and T. Previous work had polytrees (ADG+23) and Gaussian covariance constraints; this paper covers all DAGs and shows the Gaussian local Markov constraints are a sub-block. That is a real result, not a renaming of earlier equations.

The proof structure is right. Necessity (Lemma 3.5) uses independence between ε_v and proper non-descendants to get a left null vector; sufficiency (Lemma 3.6) reconstructs Λ from the unique row combination and verifies that the residual covariance and third-order tensor are diagonal. I checked the two small issues the stress-test flagged: the expansion in Lemma 3.6 writes u∈pa(w) where pa(v) is meant, and the zero-third-moment argument silently uses E[X_w]=0 when z is a descendant of v. Both are cosmetic; the repair is immediate and doesn't touch the theorem. Section 4's derivation of polytree equations from Proposition 4.2 is a nice unification, and Corollary 4.4 explains the WD19 source-recovery check as a special case.

Soft spots, in proportion:

- The numerical experiments in Section 6 are illustrative. No code or data is shipped, the graphs are tiny, and the ROC analysis is a threshold sweep on simulated data. That is fine for a theory paper, but claims about practical sink recovery should be read as suggestive, not evaluated. - The vanishing ideal is left as a conjecture, and the paper says so explicitly. Not a flaw; just a boundary. - The theorem depends on the LiNGAM error assumptions. If errors are dependent or there is unobserved confounding, the moment equations break and the rank condition loses its graph-theoretic reading. The paper is upfront about the model, and this limitation is inherent to the setting, not a gap in the proof.

Citation pattern looks normal: prior work is credited, and self-citations are to the relevant polytree and moment-variety papers. No red flag.

I would send this to a serious referee. It proves what it claims and closes a question that was explicitly open. My recommendation: accept after minor revision, with the typos in Lemma 3.6 fixed and the experiments labeled as illustrative. I would cite it.

Referee Report

1 major / 5 minor

Summary. This paper studies the set of pairs (S,T) of covariance matrices and third-order moment tensors arising from linear non-Gaussian acyclic models (LiNGAM) with a given DAG G. The main result (Theorem 3.1) states that (S,T) belongs to the model M≤3(G) if and only if for every vertex v, the matrix M_v defined in equation (2), built from S and T blocks involving parents and non-descendants of v, has rank exactly |pa(v)|. The necessity argument uses the independence of the error at v from all non-descendant variables; the sufficiency argument constructs the coefficient matrix Λ from the row dependencies of M_v and verifies that the transformed residual second- and third-order moments are diagonal. The paper also derives additional equations using a generalized parent/non-descendant operation (Proposition 4.2), recovers prior polytree results, and presents an application to sink detection.

Significance. Theorem 3.1, if correct, provides the first complete algebraic characterization of the second- and third-order moment model of LiNGAM for an arbitrary DAG, extending the polytree case of ADG+23 and containing the Gaussian local Markov rank constraints. The proof is self-contained, uses only standard linear algebra, and the constraints are explicit determinantal conditions that can be directly computed from moments. The paper includes symbolic verification in Macaulay2 for small graphs and reproducible simulations. However, the application section contains a false mathematical claim about sink recovery, overstating what Theorem 3.1 implies for structure learning.

major comments (1)

[Section 6, equation (11)] The sentence 'By Theorem 3.1, the true matrix M_i will drop rank if and only if i is not an ancestor of any other vertices in the graph' is false. Counterexample: let G be the complete DAG on {1,2,3} with edges 1→2, 1→3, 2→3, and let X1=ε1, X2=ε1+ε2, X3=ε1+ε2+ε3, where ε_i are i.i.d. with mean 0, variance 1 and third moment 1. For i=2, which is an ancestor of 3, the matrix in (11) has rows indexed by 1,3,2; direct computation gives row2 = 0.5 row1 + 0.5 row3 for every column, so its rank is 2 < 3. Thus a non-sink can produce a rank-deficient matrix, contradicting the claimed 'only if' direction. This does not affect the validity of Theorem 3.1, but it means the sink-recovery algorithm is not justified as stated; the authors should replace the claim with the correct one-directional statement (a sink implies rank deficiency) and present the method as a heuristic, or prove a different characterization.

minor comments (5)

[Section 5, proof of Lemma 3.6] In the displayed expansions of Ω^(2) and Ω^(3), the inner summation index is written as u∈pa(w) but should be u∈pa(v); otherwise the equations are inconsistent with the row relations.
[Lemma 3.5] The step 'A similar equation can be derived for t_{vwz}' implicitly uses E[ε_v X_w X_z]=0, which holds because E[X_w]=0 and ε_v is independent of X_w; this justification should be added.
[Section 2.1 and Section 6] The text says 'without loss of generality' errors have mean zero, but the simulations in Section 6 sample errors from Γ(5,1), which has mean 5; if the errors are not centered, E[X_i]≠0. Please clarify that the sampled errors are centered before use.
[Typos] There are several typos: 'knwon' in Section 2.2, 'poof' in Remark 5.1, and 'Learninig' in reference [SHHK06].
[Figure 3 caption] The caption 'The graph is a bar chart, each bar represents 200 data points' is unclear; please describe how the bar chart encodes sample sizes and proportions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.1 is derived self-contained from the LiNGAM moment parametrization.

full rationale

The central claim, Theorem 3.1, is not circular. Necessity (Lemma 3.5) follows directly from the structural equations and the LiNGAM error-independence assumption: for w a non-descendant of v, E[ε_v X_w]=0, which yields the row relations (7) and (8); together with positive definiteness of S this gives rank exactly |pa(v)|. Sufficiency (Lemma 3.6) is constructive: the rank condition makes the bottom row of each M_v a linear combination of the parent rows, and the coefficients define Λ∈R^E; direct calculation from the definitions (4) and (5) shows Ω^(2) and Ω^(3) become diagonal. No fitted data value or target membership is used as an input to characterize M≤3(G). The paper cites prior work by overlapping authors ([ADG+23], [RS21], [SRD24]), but only for context, for the standard moment transformation formula, and as a special case to recover; the polytree characterization is re-derived from Proposition 4.2 rather than assumed. The unsolved vanishing ideal issue is explicitly flagged as conjectural/future work and is not used as a premise. The dependence on error independence is a modeling assumption of LiNGAM, not a circular step.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The theorem is a pure algebraic characterization; the only inputs are the LiNGAM moment parametrization and standard linear algebra. No free parameters are fitted and no new entities are postulated. The load-bearing domain assumption is mutual independence of the errors, without which the moment relations in Lemma 3.5 fail.

assumptions (3)

domain assumption The errors ε_i are mutually independent, have mean zero, positive finite variances, and finite third moments.
This is the LiNGAM setup in Section 2.1; Lemma 3.5 uses the stronger consequence that ε_v is independent of every proper non-descendant X_w, producing the rank relations (7) and (8).
standard math The matrix I - Λ is invertible for any DAG, and Λ is nilpotent under a topological ordering of the vertices.
Needed in Section 2.1 to write X = (I-Λ)^-T ε and to guarantee that the Λ constructed in Lemma 3.6 yields a positive definite Ω(2); this follows from acyclicity.
standard math Positive definiteness of S is inherited by the submatrix S_pa(v),pa(v), so the row relation for M_v determines unique parent coefficients λ_v.
Used in Lemma 3.6 and Remark 3.2; this is why the rank condition is equality with |pa(v)| rather than an inequality.

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Cite this review

Pith. "Pith review of Algebraic Constraints for Linear Acyclic Causal Models." pith.science (2026). https://pith.science/paper/6CUABHZS

@misc{pith2026250500215,
  author       = {Pith},
  title        = {Pith review of: Algebraic Constraints for Linear Acyclic Causal Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CUABHZS}},
  note         = {Machine review of arXiv:2505.00215}
}

abstract

In this paper we study the space of second- and third-order moment tensors of random vectors which satisfy a Linear Non-Gaussian Acyclic Model (LiNGAM). In such a causal model each entry $X_i$ of the random vector $X$ corresponds to a vertex $i$ of a directed acyclic graph $G$ and can be expressed as a linear combination of its direct causes $\{X_j: j\to i\}$ and random noise. For any directed acyclic graph $G$, we show that a random vector $X$ arises from a LiNGAM with graph $G$ if and only if certain easy-to-construct matrices, whose entries are second- and third-order moments of $X$, drop rank. This determinantal characterization extends previous results proven for polytrees and generalizes the well-known local Markov property for Gaussian models.

Figures

Figures reproduced from arXiv: 2505.00215 by the authors.

**Figure 1.** The complete DAG on 3 vertices. Example 1.1. Consider the graph G with vertices V = {1, 2, 3} and edges {1 → 2, 1 → 3, 2 → 3}, the simplest DAG which is not a polytree (see [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** The complete DAG on 5 vertices [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** At each number of samples n, we record the number of times each vertex was assigned as the sink out of the entire data set. The graph is a bar chart, each bar represents 200 data points. 1 2 3 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The line DAG on 3 vertices 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: True positive rate plotted against the false positive rate for different threshold [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages

[1]

Third-order moment varieties of linear non-Gaussian graphical models

Carlos Am\' e ndola, Mathias Drton, Alexandros Grosdos, Roser Homs, and Elina Robeva. Third-order moment varieties of linear non-Gaussian graphical models . Information and Inference: A Journal of the IMA , 12(3):1405--1436, 04 2023

work page 2023
[2]

Real birational implicitization for statistical models, 2024

Tobias Boege and Liam Solus. Real birational implicitization for statistical models, 2024

work page 2024
[3]

Symmetric tensors and symmetric tensor rank

Pierre Comon, Gene Golub, Lek-Heng Lim, and Bernard Mourrain. Symmetric tensors and symmetric tensor rank. SIAM Journal on Matrix Analysis and Applications , 30(3):1254--1279, 2008

work page 2008
[4]

Mathias Drton, Marina Garrote-Lop\'ez, Niko Nikov, Elina Robeva, and Samuel Y. Wang. Causal discovery for linear non-gaussian models with disjoint cycles. submitted to UAI 2025 , 2025

work page 2025
[5]

Structure learning in graphical modeling

Mathias Drton and Marloes Maathuis. Structure learning in graphical modeling. Annual Review of Statistics and Its Application , 4:365--393, 2017

work page 2017
[6]

Nested covariance determinants and restricted trek separation in gaussian graphical models

Mathias Drton, Elina Robeva, and Luca Weihs. Nested covariance determinants and restricted trek separation in gaussian graphical models. Bernoulli , 26(4):2503--2540, 2020

work page 2020
[7]

Positivity for gaussian graphical models

Jan Draisma, Seth Sullivant, , and Kelli Talaska. Positivity for gaussian graphical models. Adv. in Appl. Math. , 50(5):661--674, 2013

work page 2013
[8]

Kolda and B

T. Kolda and B. Badre. Tensor decompositions and applications. SIAM Review , 51(3):455--500, 2009

work page 2009

Show all 19 references

[9]

Handbook of graphical models

Marloes Maathuis, Mathias Drton, Steffen Lauritzen, and Martin Wainwright. Handbook of graphical models . CRC Press, 2018

2018
[10]

Multi-trek separation in linear structural equation models

Elina Robeva and Jean-Baptiste Seby. Multi-trek separation in linear structural equation models. SIAM Journal on Applied Algebra and Geometry , 5(2):278--303, 2021

2021
[11]

Goodness-of-fit tests for linear non-gaussian structural equation models, 2023

Daniela Schkoda and Mathias Drton. Goodness-of-fit tests for linear non-gaussian structural equation models, 2023

2023
[12]

arinen, and Antti

Shohei Shimizu, Patric. O. Hoyer, Aapo Hyv\"arinen, and Antti" Kerminen. A linear non-gaussian acyclic model for causal discovery. Journal of Machine Learninig Research , 7:2003--2030, 2006

2003
[13]

Causal discovery of linear non-gaussian causal models with unobserved confounding

Daniela Schkoda, Elina Robeva, and Mathias Drton. Causal discovery of linear non-gaussian causal models with unobserved confounding. arXiv:2408.04907 , 2024

2024 arXiv
[14]

Trek separation for gaussian graphical models

Seth Sullivant, Kelli Talaska, and Jan Draisma. Trek separation for gaussian graphical models. Annals of Statistics , 38(3):1665--1685, 2010

2010
[15]

Algebraic geometry of gaussian bayesian networks

Seth Sullivant. Algebraic geometry of gaussian bayesian networks. Advances in Applied Mathematics , 40(4):482--513, 2008

2008
[16]

Algebraic Statistics , volume 194 of Graduate Studies in Mathematics

Seth Sullivant. Algebraic Statistics , volume 194 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2018

2018
[17]

Thijs van Ommen and Joris M. Mooij. Algebraic equivalence of linear structural equation models. In Proceedings of the 33rd Annual Conference on Uncertainty in Artificial Intelligence (UAI-17) , 2017

2017
[18]

High-dimensional causal discovery under non-Gaussianity

Y Samuel Wang and Mathias Drton. High-dimensional causal discovery under non-Gaussianity . Biometrika , 107(1):41--59, 10 2019

2019
[19]

Samuel Wang and Mathias Drton

Y. Samuel Wang and Mathias Drton. Causal discovery with unobserved confounding and non- G aussian data. J. Mach. Learn. Res. , 24:Paper No. [271], 61, 2023

2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Third-order moment varieties of linear non-Gaussian graphical models

Carlos Am\' e ndola, Mathias Drton, Alexandros Grosdos, Roser Homs, and Elina Robeva. Third-order moment varieties of linear non-Gaussian graphical models . Information and Inference: A Journal of the IMA , 12(3):1405--1436, 04 2023

work page 2023

[2] [2]

Real birational implicitization for statistical models, 2024

Tobias Boege and Liam Solus. Real birational implicitization for statistical models, 2024

work page 2024

[3] [3]

Symmetric tensors and symmetric tensor rank

Pierre Comon, Gene Golub, Lek-Heng Lim, and Bernard Mourrain. Symmetric tensors and symmetric tensor rank. SIAM Journal on Matrix Analysis and Applications , 30(3):1254--1279, 2008

work page 2008

[4] [4]

Mathias Drton, Marina Garrote-Lop\'ez, Niko Nikov, Elina Robeva, and Samuel Y. Wang. Causal discovery for linear non-gaussian models with disjoint cycles. submitted to UAI 2025 , 2025

work page 2025

[5] [5]

Structure learning in graphical modeling

Mathias Drton and Marloes Maathuis. Structure learning in graphical modeling. Annual Review of Statistics and Its Application , 4:365--393, 2017

work page 2017

[6] [6]

Nested covariance determinants and restricted trek separation in gaussian graphical models

Mathias Drton, Elina Robeva, and Luca Weihs. Nested covariance determinants and restricted trek separation in gaussian graphical models. Bernoulli , 26(4):2503--2540, 2020

work page 2020

[7] [7]

Positivity for gaussian graphical models

Jan Draisma, Seth Sullivant, , and Kelli Talaska. Positivity for gaussian graphical models. Adv. in Appl. Math. , 50(5):661--674, 2013

work page 2013

[8] [8]

Kolda and B

T. Kolda and B. Badre. Tensor decompositions and applications. SIAM Review , 51(3):455--500, 2009

work page 2009