Writing study: The Spacetime of Diffusion Models: An Information Geometry Perspective

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← ICLR 2026 oral writing corpus

Rafał Karczewski, Markus Heinonen, Alison Pouplin, Søren Hauberg, Vikas Garg · 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

25 physical pages

Before references / appendix

≈ 9.36 page-equivalents

Main figures / tables

7 / 1

Attached appendix figures / tables

1 / 0

Attached appendix span

≈ 13.62 page-equivalents

Before references / appendix: ≈ 37.44%References / intervening material: ≈ 8.08%Attached appendix: ≈ 54.48%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	≈ 0.78 pp.	3.12%
2 BACKGROUND ON DIFFUSION MODELS	2	≈ 0.3 pp.	1.2%
3 RIEMANNIAN GEOMETRY OF DIFFUSION MODELS	2	≈ 0.76 pp.	3.04%
4 PULLBACK GEOMETRY COLLAPSES IN DIFFUSION MODELS	3	≈ 0.44 pp.	1.76%
5 INFORMATION GEOMETRY WITH DENOISING DECODERS	3	≈ 1.75 pp.	7%
6 EXPERIMENTS	5	≈ 3.16 pp.	12.64%
7 RELATED WORKS	8	≈ 0.69 pp.	2.76%
8 LIMITATIONS	9	≈ 0.18 pp.	0.72%
9 CONCLUSION	9	≈ 0.17 pp.	0.68%
REPRODUCIBILITY STATEMENT	9	≈ 0.07 pp.	0.28%
ETHICS STATEMENT	9	≈ 0.36 pp.	1.44%
ACKNOWLEDGMENTS	10	≈ 0.13 pp.	0.52%
REFERENCES	10	≈ 2.02 pp.	8.08%
A NOTATION	12	≈ 0.73 pp.	2.92%
B PULLBACK GEOMETRY IN DIFFUSION MODELS	13	≈ 1.17 pp.	4.68%
C PROOF OF PROPOSITION 5.1	14	≈ 3.89 pp.	15.56%
D KULLBACK-LEIBLER DIVERGENCE IN EXPONENTIAL FAMILIES	18	≈ 2.94 pp.	11.76%
E GEODESIC AND DIFFED PSEUDOCODE	21	≈ 0.58 pp.	2.32%
F QUALITATIVE EXAMPLE OF DIFFED	21	≈ 0.12 pp.	0.48%
G EXPERIMENTAL DETAILS	21	≈ 2.74 pp.	10.96%
H NOTE ON TRANSITION PATH SAMPLING BASELINES	24	≈ 0.35 pp.	1.4%
I EXPECTATION PARAMETER ESTIMATION CODE	24	≈ 0.51 pp.	2.04%
J LICENCES	25	≈ 0.59 pp.	2.36%

All detected figure and table locations

Main text

• Figure 1 · p. 1
• Figure 2 · p. 3
• Figure 3 · p. 5
• Figure 4 · p. 5
• Figure 5 · p. 6
• Figure 6 · p. 7
• Table 1 · p. 7
• Figure 7 · p. 9

Attached appendix

• Figure 8 · p. 22

Argument and organization

1. Two geometrical failures motivate spacetime

   Rules out a full-dimensional Euclidean pullback and terminal-time information geometry before introducing noisy-state/time coordinates.

2. Tractable estimation, applications and limits

   Derives moment-based energy estimates, demonstrates interpolation and transition paths, then specifies computational and endpoint limitations.

Problem, claim and contribution

The introduction establishes a precise obstruction instead of announcing a new metric without explaining why familiar constructions fail.

Sections 1 and 4–5, pp. 1–4. · Main paper

Argument and section order

A boxed computational takeaway bridges the exponential-family derivation to an implementation using a pretrained denoiser and Jacobian-vector products.

Section 5, p. 4; Appendix C. · Main paper

Figures, captions and visual explanation

The compact noise/data-space diagram contrasts a curved latent path with its straight decoded path; the caption states the mathematical conclusion rather than merely naming the drawing.

Figure 2, p. 3. · Main paper

Tables, comparisons and evidence

Transition-path visualizations and a cost/quality table complement each other. The table explicitly separates the additional 16M training-data evaluations from the 16M path-generation evaluations.

Figure 6 and Table 1, p. 7. · Main paper

Appendix and reproducibility

The supplement gives optimization settings, runtimes, code and a candid account of unavailable or difficult-to-reproduce baselines, materially qualifying the comparison.

Appendices E–J, pp. 21–25. · Main paper

Clarity and writing conventions

The limitations distinguish a computable geometric quantity from a fast metric or standalone sampler: fixed endpoints and substantial optimization remain necessary.

Section 8, p. 10; Appendix G.2, pp. 22–23. · Main paper

What to reuse—and what not to infer

Transferable lesson: Build a theory-driven method as obstruction → revised mathematical object → tractable estimator → task-specific evidence, and place cost accounting beside quality.

Boundary: The pullback critique assumes a full-dimensional bijective decoder; it is not a general rejection of manifold geometry. Image-distance correlations are not human-perception validation, and molecular results use a two-angle surrogate with documented baseline difficulties.

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 1–8, including equations, algorithms, experiments and limitations.

Appendix / supplement

Pages 13–25, Appendices A–J: notation, proofs, estimators, pseudocode, experiments, baseline reproducibility and licenses.

Visually inspected PDF pages

3, 7

• ICLR 2026 official proceedings PDF, retrieved 2026-09-19 · 25 pages