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REVIEW 2 major objections 6 minor 34 references

On generating random Gaussian graphical models

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For chordal graphs, correlation matrices respecting the graph's zero pattern can be drawn uniformly, and a hybrid of uniform sampling and orthogonalization extends the construction to general graphs.

desk verdict A useful, honest methods paper that gives chordal uniform sampling a real try; the main practical claim is only as good as the unexamined Metropolis chain underneath it. read the letter →

arxiv 1909.01062 v2 pith:2P4CMY7T submitted 2019-09-03 stat.ME

Irene Córdoba , Gherardo Varando , Concha Bielza , Pedro Larrañaga This is my paper · ORCID
classification stat.ME
keywords randomcorrelationmatrixchordalgraphCholeskyparametrizationMetropolis-HastingssamplingpartialorthogonalizationGaussiangraphicalmodelstructurelearningvalidationpositivedefinitesimulation
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 addresses how synthetic Gaussian graphical models are built for validating structure-learning algorithms. It argues that the usual trick of enforcing a dominant diagonal biases the generated matrices toward tiny off-diagonal entries, and shows experimentally that this bias changes which algorithms appear to work. The main positive result is that when the underlying undirected graph is chordal (every cycle of length four or more has a chord), one can sample uniformly from the set of correlation matrices compatible with that graph, using a row-wise Cholesky parametrization and a Metropolis step on hemispheres. For non-chordal graphs, the authors triangulate the graph, sample uniformly on the triangulation, and then apply partial orthogonalization to delete the added edges, producing a matrix in the desired set whose entries explore the off-diagonal range more fully.

What carries the argument

The load-bearing object is the Cholesky parametrization $\Phi: U \mapsto UU^t$ restricted to $U^p_1(G)$, the set of unit-row upper-triangular matrices whose zero pattern matches the missing edges of $G$; for chordal $G$ this map is a bijection onto $E_p(G)$ and its Jacobian factorizes across rows, reducing uniform sampling on the elliptope to independent one-dimensional-hemisphere Metropolis draws. The second mechanism is modified Gram-Schmidt orthogonalization of each row against the rows it must be orthogonal to, which enforces the zero pattern of $E_p(G)$ and serves both as a standalone generator and as the cleanup step that removes fill-in edges after triangulation.

What would settle it

Run Algorithm 3 on a chordal graph with a high-degree node, collect many draws, and compare the empirical distribution of off-diagonal entries with the exact uniform marginals on $E_p(G)$ (available in closed form for small graphs such as the three-node chain, where $E_3(G)$ is the unit disk); a systematic discrepancy that grows with dimension or degree would falsify the uniformity claim.

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Extended reading notes

Core claim

For a chordal undirected graph $G$ with a perfect elimination ordering, the paper establishes a one-to-one parametrization of the elliptope slice $E_p(G)$ by unit-row upper-triangular Cholesky factors: $\Phi(U)=UU^t$ with $U \in U^p_1(G)$. The Jacobian of this map, $\det(J_\Phi(U))=2^p \prod_{i=1}^p u_{ii}^{|\operatorname{pa}(i)|+1}$, factorizes row by row, so the uniform measure on $E_p(G)$ pulls back to independent distributions on hemispheres $S_+^{|\mathrm{ch}(i)|}$, each with density proportional to the first coordinate raised to $|\operatorname{pa}(i)|+1$. The paper then samples those rows with a Metropolis-Hastings step and maps back through $\Phi$, giving a uniform draw from $E_p(G)$. For a general undirected graph, the paper's Algorithm 5 samples a Cholesky factor uniformly on a triangulation and then orthogonalizes rows against non-neighbours, returning a matrix in $E_p(G)$ without a stated target distribution.

Load-bearing premise

The uniformity guarantee depends on the random-walk sampling step having converged to its target distribution by the time the algorithm stops, for every graph size and node degree used; no mixing analysis, tuning values, or diagnostics are supplied, and for the non-chordal hybrid the output distribution is not described at all.

Editorial extensions

If this is right

  • For chordal graphs, Algorithm 3 gives a uniform sampler over $E_p(G)$ (up to Markov-chain error), so validation studies can benchmark against the full set of compatible correlation matrices instead of a biased subset.
  • Structure-learning comparisons built on diagonally dominant matrices can mis-rank methods: the paper's replication of an earlier benchmark study shows rankings change when the proposed generator is used, especially at higher graph density.
  • The row-wise factorization means the computational cost of the uniform sampler is governed by parent and child counts in a perfect elimination ordering, not by the full $p \times p$ matrix, so the method remains usable for moderately large chordal graphs.
  • For non-chordal graphs, the hybrid method guarantees a matrix in $E_p(G)$ and spreads off-diagonal entries more than diagonal dominance, though it does not carry a uniformity guarantee.

Reading between the lines

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

  • The paper leaves the Metropolis step uncharacterized; a natural calibration exercise would be to compare Algorithm 3's output with exact uniform marginals on small chordal graphs (e.g., the three-node chain, where $E_3(G)$ is the unit disk) to pick burn-in and noise variance.
  • Because Algorithm 5's output depends on the chosen triangulation and vertex ordering, one could randomize over triangulations or average over orderings to obtain a better-defined reference distribution for non-chordal graphs.
  • The same Cholesky-row sampling scheme transfers directly to concentration graphs, since $\Omega = UU^t$ with zeros in the upper triangle is exactly the decomposition the paper uses; the validation comparisons shown are for correlation/covariance matrices, but the generator applies to the precision-matrix case as well.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The manuscript proposes methods for generating symmetric positive definite correlation matrices whose zero pattern is prescribed by an undirected graph, with the aim of improving synthetic-model generation for validating structure learning algorithms. For chordal graphs, it develops a uniform sampler over the set E_p(G) of compatible correlation matrices. The method uses the Cholesky parametrization U ↦ UU^t, whose Jacobian factorizes across rows, and samples each row independently on a hemisphere S_+^α from a density proportional to v_1^γ with γ = |pa(i)| + 1, using the Metropolis-Hastings algorithm mh_u. For non-chordal graphs, the paper proposes a hybrid method: sample uniformly from a triangulation of G and then apply partial orthogonalization to remove the edges added by the triangulation. The experiments examine marginal distributions of matrix entries for the different generators and compare rankings of structure learning algorithms under diagonal dominance versus the proposed hybrid method.

Significance. Conditional on the Metropolis step converging, the row-factorization argument in Section 3.3 is clean and gives a principled way to sample uniformly from E_p(G) for chordal graphs, which is a useful alternative to the prevailing diagonal-dominance constructions. The paper also ships an R package and reproduction code, which is a concrete asset. The three-variable example (Figure 2) is a convincing sanity check of the uniform target. The empirical finding that the matrix generator changes the ranking of structure learning algorithms is practically relevant. The principal weakness is that the uniformity claim relies on a finite-run Metropolis sampler whose tuning parameters and convergence behavior are not documented; this is fixable but currently load-bearing.

major comments (2)
  1. [Section 3.3, Algorithms 3-4, Eq. (9)] The claim that Algorithm 3 samples uniformly from E_p(G) for chordal G holds only if mh_u returns a draw from f(v) ∝ v_1^γ on S_+^α. Algorithm 4 is a finite-run Metropolis chain whose two free parameters, the proposal variance σ_epsilon and the burn-in t_b, are never assigned numerical values anywhere in the manuscript, and no acceptance rates, autocorrelation times, or other convergence diagnostics are reported. In Section 4.2, γ can be around 50, where the target density is sharply concentrated near v_1 = 1 and a poorly scaled Gaussian proposal can mix very slowly. Because the uniform-sampling claim is the central contribution, this is a load-bearing gap. Please report the values of σ_epsilon and t_b used in all experiments and provide convergence diagnostics, or alternatively present the result explicitly as an asymptotic statement with a practical verification protocol.
  2. [Section 3.4, Algorithm 5 and Figures 3-6] For non-chordal graphs, the output of Algorithm 5 depends on the choice of triangulation and of the permutation σ, but the paper does not discuss this dependence or characterize the induced distribution on E_p(G). The figure labels such as "Uniform (+ partial orthogonalization)" can therefore be misleading, because the final output is not uniform over E_p(G). The paper should state explicitly that this hybrid is a heuristic whose distribution is unknown and that different triangulations or permutations may lead to different generators.
minor comments (6)
  1. [Section 2.3] The sentence "In the Gaussian case, this implies that if 1≺···≺ p is a perfect ordering for G." is incomplete; please finish it, for example by stating that S^p_{>0}(G) = B(G_D) under a perfect ordering.
  2. [Algorithm 3] The notation in Algorithm 3 for assigning the non-first entries of v to the children columns is garbled in the text; it should read u_{i,ch(i)} = v_{-1}, where v_{-1} is the vector v without its first entry.
  3. [Section 4.3] The text says "there is a significant improvement in the densest (d = 0.5)", but Figure 7 is labeled d = 0.25; please correct the value or the figure caption to be consistent.
  4. [Algorithm 4 and Section 3.3] The comment "with default noise variance σ_epsilon and burn-in time t_b" implies that defaults exist, but no values are given; if the defaults come from Córdoba et al. (2018), please provide the values or an explicit pointer.
  5. [Figures 3, 4, and 6] The captions say "marginal densities of the non-zero entries", but the plots only appear to show the first and last non-zero entries; please state explicitly which entries are displayed.
  6. [References] The two entries by Córdoba et al. from 2018 are not distinguished in citations; please label them 2018a and 2018b to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chordal uniform-sampling claim follows from a standard area-formula argument, and the Metropolis sampler is fully specified with the correct stationary distribution.

full rationale

The central claim that Algorithm 3 samples uniformly from Ep(G) for chordal G is derived, not assumed. The paper parametrizes Ep(G) by unit-row Cholesky factors with the graph zero pattern, uses the known Jacobian det(JPhi(U)) = 2^p times the product of u_ii^{pa(i)+1} (Eq. 9), and then applies the area formula of Diaconis et al. (2013) to reduce uniform sampling on Ep(G) to independent row sampling from a density proportional to v_1^gamma on S_+^alpha. Each row of U is sampled by the Metropolis-Hastings algorithm in Algorithm 4, whose acceptance ratio (v'_1/v_t1)^gamma has the stated target f(v) proportional to v_1^gamma as its stationary distribution. No fitted parameter is renamed as a prediction, and no data subset is used to calibrate the result. The citations to Cordoba et al. (2018) provide the earlier Metropolis procedure and an alternative statement of the Jacobian, but the algorithm is reproduced in full in the paper and the Jacobian is also attributed to Roverato (2000); these are normal prior-work references rather than load-bearing self-citations. The non-chordal case is explicitly not claimed to be uniform: Algorithm 5 combines uniform sampling on a triangulation with partial orthogonalization, and the conclusions state that sampling uniformly from Ep(G) for non-chordal G remains future work. The absence of mixing diagnostics or explicit tuning values for sigma_epsilon and t_b is a practical convergence concern, not a circularity, because the target distribution of the Metropolis chain is correctly specified by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper's central contribution rests on a published Jacobian, the authors' own MH sampler, and an uncharacterized hybrid scheme for non-chordal graphs. No new physical or mathematical entities are introduced.

free parameters (2)
  • MH noise variance sigma_epsilon
    Used in Algorithm 4's proposal step; convergence and acceptance probability depend on it, but no value or tuning guidance is given in the paper.
  • MH burn-in t_b
    Number of iterations discarded before returning the sample; no value is provided, and there is no convergence diagnostic to justify the choice.
assumptions (4)
  • standard math The Cholesky factorization parametrization Phi: U_p^1(G) -> E_p(G) is one-to-one for chordal graphs with a perfect ordering (Section 2.3, Eq. 8).
    This equivalence between decomposable models and DAGs with no v-structures is established in the cited literature (Wermuth 1980; Paulsen et al. 1989) and assumed without proof here.
  • standard math The Jacobian determinant det(J_Phi(U)) = 2^p * prod_i u_ii^{|pa(i)|+1} (Eq. 9) holds for the Cholesky parametrization.
    Attributed to Roverato (2000) and Córdoba et al. (2018); used to define the row densities for the MH sampler. The paper relies on this result rather than deriving it.
  • ad hoc to paper The Metropolis-Hastings sampler mh_u (Algorithm 4) samples from f(v) proportional to v_1^gamma on the positive hemisphere S_+^alpha.
    The sampler was introduced in the authors' own Córdoba et al. (2018) IDEAL paper and is not proven to converge here; the paper provides no mixing or error analysis and leaves tuning parameters unspecified.
  • domain assumption For non-chordal graphs, applying uniform sampling to a triangulation and then partial orthogonalizing to remove triangulation edges yields matrices whose distribution is a useful proxy for uniform over E_p(G).
    The paper does not characterize this distribution; it only demonstrates empirically in Section 4 that marginal densities look reasonable and that validation rankings change. Whether this distribution is adequate for validation is assumed rather than proven.

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Cite this review

Pith. "Pith review of On generating random Gaussian graphical models." pith.science (2026). https://pith.science/paper/2P4CMY7T

@misc{pith2026190901062,
  author       = {Pith},
  title        = {Pith review of: On generating random Gaussian graphical models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2P4CMY7T}},
  note         = {Machine review of arXiv:1909.01062}
}
read the original abstract

Structure learning methods for covariance and concentration graphs are often validated on synthetic models, usually obtained by randomly generating: (i) an undirected graph, and (ii) a compatible symmetric positive definite (SPD) matrix. In order to ensure positive definiteness in (ii), a dominant diagonal is usually imposed. In this work we investigate different methods to generate random symmetric positive definite matrices with undirected graphical constraints. We show that if the graph is chordal it is possible to sample uniformly from the set of correlation matrices compatible with the graph, while for general undirected graphs we rely on a partial orthogonalization method.

Figures

Figures reproduced from arXiv: 1909.01062 by the authors.

Figure 1
Figure 1. Chordal undirected graph with three variables. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Scatter plot of the two non-zero entries for correlation matrices sampled from [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Marginal densities of the non-zero entries of matrices sampled from [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Marginal densities of the non-zero entries of matrices sampled from [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Chordal undirected graph Gchain with 50 variables and 49 edges 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Marginal densities of the non-zero entries of matrices sampled from [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: True positive rate (TPR) and true discovery rate (PPV) of the structure learning algorithms [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: True positive rate (TPR) and true discovery rate (PPV) of the structure learning algorithms [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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