Writing study: Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning

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← ICLR 2026 oral writing corpus

Haoyue Dai, Immanuel Albrecht, Peter Spirtes, Kun Zhang · Official ICLR 2026 oral record

Observed writing choices, not acceptance rules. This study examines the inspected manuscript version. It does not establish why the paper was accepted or that its presentation choices should be copied unchanged.

Figure, table and section ledger

Editorial contribution type: Theory & formal foundations (primary) · Learning & optimization algorithms.

Layout extraction, not a required paper template. Counts are distinct caption labels; a multi-panel figure counts once. Section spans are estimated from heading positions, including floats and whitespace. Method and limitations.

Canonical PDF

48 physical pages

Before references / appendix

≈ 10.34 page-equivalents

Main figures / tables

3 / 0

Attached appendix figures / tables

5 / 5

Attached appendix span

≈ 29.87 page-equivalents

Before references / appendix: ≈ 21.54%References / intervening material: ≈ 16.23%Attached appendix: ≈ 62.23%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Section-by-section lengths and source pages

The title/abstract/front matter precede the first numbered heading. A span ends at the next detected top-level heading, not at the last sentence; end-page whitespace is included. References and statements are listed separately. These are not measurements of text-only or artwork area.

Top-level heading	Starts on PDF page	Estimated span	Share of entire PDF
1 INTRODUCTION	1	≈ 1.32 pp.	2.75%
2 PROBLEM SETUP	2	≈ 1.84 pp.	3.83%
3 DEVELOPING GRAPHICAL TOOLS FOR CHARACTERIZING EQUIVALENCE	4	≈ 3.06 pp.	6.38%
4 THE GRAPHICAL CHARACTERIZATION OF DISTRIBUTIONAL EQUIVALENCE	7	≈ 2 pp.	4.17%
5 ALGORITHM AND EVALUATION	9	≈ 1.06 pp.	2.21%
6 CONCLUSION AND LIMITATIONS	10	≈ 0.52 pp.	1.08%
REFERENCES	11	≈ 7.79 pp.	16.23%
Appendix	19	≈ 0.55 pp.	1.15%
A DETAILS OF THE GLVLING ALGORITHM	19	≈ 4.92 pp.	10.25%
B PROOFS OF MAIN RESULTS	24	≈ 11.5 pp.	23.96%
C DISCUSSION	36	≈ 7 pp.	14.58%
D EVALUATION RESULTS	43	≈ 5.89 pp.	12.27%

All detected figure and table locations

Main text

• Figure 1 · p. 4
• Figure 2 · p. 6
• Figure 3 · p. 9

Attached appendix

• Figure 4 · p. 31
• Table 1 · p. 36
• Figure 5 · p. 36
• Figure 6 · p. 37
• Table 2 · p. 40
• Table 3 · p. 43
• Table 4 · p. 44
• Table 5 · p. 45
• Figure 7 · p. 47
• Figure 8 · p. 48

Argument and organization

1. Introduction and preliminaries

   Frame identifiability before algorithm design and distinguish distributional from Markov equivalence.

2. Equivalence characterization and learning

   Develop path ranks, local edge ranks, graphical and transformational criteria, then the glvLiNG algorithm.

3. Appendices A–D

   Separate implementation/proofs, conceptual extensions, and extensive evaluation.

Problem, claim and contribution

The introduction uses three explicit questions to connect the identifiability gap, equivalence criteria, and learning algorithm. Familiar CPDAG and covered-edge-reversal analogies make the new setting legible without equating the two theories.

Sections 1–2, pp. 1–4. · Main paper

Argument and section order

Definitions and propositions build a dependency chain from non-Gaussian mixing identifiability to path ranks and then local edge ranks; the algorithm appears after the equivalence characterization, not before its target is defined.

Sections 3–4, pp. 4–10. · Main paper

Figures, captions and visual explanation

Figure 2 reuses orange, red, and blue vertex groups across a directed graph, bipartite matching, and support matrix. Its caption computes both ranks on the same example, turning an abstract duality into a worked visual argument.

Figure 2, p. 6. · Main paper

Tables, comparisons and evidence

Figure 7 uses a shared legend and matrix of panels for graph size, latent count, density, and sample size, with mean and standard deviation over 25 models. The accompanying text acknowledges that competing methods can win in sparse settings.

Appendix D.4, pp. 46–47, Figure 7. · Main paper

Appendix and reproducibility

The supplement layers a notation/matroid tutorial, dependency-ordered proofs, an equivalence-class presentation, and evaluations. Table 4 explicitly labels oracle-input runtime, while D.4 separately describes empirical-rank confidence scores and latent-count selection.

Appendices A–D, pp. 19–48; Table 4, p. 44; D.4, p. 46. · Main paper

What to reuse—and what not to infer

Transferable lesson: Use a recurring small graph to connect algebra, graph operations, and identifiability; report oracle guarantees and finite-sample evidence as distinct claims.

Boundary: Structural-assumption-free does not mean assumption-free: the model is linear with independent non-Gaussian errors, and the learning guarantee relies on oracle mixing/rank conditions. The observational stock example has no intervention-based ground truth; Appendix D is essential because the main text gives only a brief evaluation overview.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-09-19

Reading status

Full text + selected visuals inspected

Main text

Pages 1–10, Sections 1–4; references pp. 11–18.

Appendix / supplement

Appendices A–D, pp. 19–48: preliminaries, algorithms, proofs, equivalence-class presentations, related work, exhaustive enumeration, oracle and finite-sample experiments, and stock-market example.

Visually inspected PDF pages

6, 47

• ICLR 2026 official proceedings PDF, retrieved 2026-09-19 · 48 pages