Writing study: On the Wasserstein Geodesic Principal Component Analysis of probability measures

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← ICLR 2026 oral writing corpus

Nina Vesseron, Elsa Cazelles, Alice Le Brigant, Klein Thierry · 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

32 physical pages

Before references / appendix

≈ 10.21 page-equivalents

Main figures / tables

7 / 0

Attached appendix figures / tables

14 / 3

Attached appendix span

≈ 18.89 page-equivalents

Before references / appendix: ≈ 31.91%References / intervening material: ≈ 9.06%Attached appendix: ≈ 59.03%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	≈ 1.09 pp.	3.41%
2 BACKGROUND	2	≈ 1.67 pp.	5.22%
3 GEODESIC PCA ON CENTERED GAUSSIAN DISTRIBUTIONS	4	≈ 1.45 pp.	4.53%
4 GEODESIC PCA ON A.C. PROBABILITY MEASURES: GPCAGEN	5	≈ 1.58 pp.	4.94%
5 EXPERIMENTS	7	≈ 3.22 pp.	10.06%
6 DISCUSSION	10	≈ 0.27 pp.	0.84%
ACKNOWLEDGEMENTS	10	≈ 0.29 pp.	0.91%
REPRODUCIBILITY STATEMENT	11	≈ 0.11 pp.	0.34%
REFERENCES	11	≈ 2.89 pp.	9.03%
A ADDITIONAL EXPERIMENTS AND FIGURES	14	≈ 5.33 pp.	16.66%
B THE OTTO-WASSERSTEIN GEOMETRY	19	≈ 3.83 pp.	11.97%
C LINEARIZED OPTIMAL TRANSPORT AND TANGENT PCA	23	≈ 1.12 pp.	3.5%
D GEODESIC PCA FOR GAUSSIAN DISTRIBUTIONS	24	≈ 4.71 pp.	14.72%
E HYPERPARAMETERS	29	≈ 3.4 pp.	10.63%
F USE OF LARGE LANGUAGE MODELS (LLMS)	32	≈ 0.5 pp.	1.56%

All detected figure and table locations

Main text

• Figure 1 · p. 3
• Figure 2 · p. 5
• Figure 3 · p. 7
• Figure 4 · p. 8
• Figure 5 · p. 8
• Figure 6 · p. 9
• Figure 7 · p. 10

Attached appendix

• Figure 8 · p. 14
• Figure 9 · p. 14
• Figure 10 · p. 14
• Figure 11 · p. 15
• Figure 12 · p. 15
• Figure 13 · p. 16
• Figure 14 · p. 16
• Figure 15 · p. 17
• Figure 16 · p. 17
• Figure 17 · p. 18
• Figure 18 · p. 19
• Figure 19 · p. 22
• Table 1 · p. 29
• Table 2 · p. 30
• Figure 20 · p. 31
• Table 3 · p. 31
• Figure 21 · p. 31

Argument and organization

1. Geometric motivation and background

   Explain why ordinary PCA on density functions is inappropriate before introducing Wasserstein geometry.

2. Gaussian case then general distributions

   Use a finite-dimensional construction as the scaffold for the neural parameterization.

3. Experiments and discussion

   Show agreement, divergence and interpretable applications instead of only superiority.

Problem, claim and contribution

The introduction distinguishes minimizing projection residuals from maximizing projected variance; this prevents a familiar PCA name from hiding a changed mathematical objective.

Sections 1–2, pp. 1–2 · Main paper

Argument and section order

The Gaussian case establishes the lifting idea before general absolutely continuous measures, with propositions separating cited geometry from new results.

Sections 2–4, pp. 2–7 · Main paper

Figures, captions and visual explanation

Figure 1 places the flat total space over the curved distribution space, with aligned fibers and a colored geodesic; its caption credits the source of the visual idea.

Figure 1, p. 3 · Main paper

Tables, comparisons and evidence

Figure 4 pairs a 3D construction, a 2D simplification and a quantitative distortion sweep; the caption identifies solid/dashed methods, while nearby prose acknowledges undesirable boundary projections.

Figure 4 and Section 5.1, p. 8 · Main paper

Appendix and reproducibility

Proofs, baseline definitions and implementation details are separate. Regularization tables report both constraint satisfaction and objective cost, rather than treating the smallest training loss as automatically correct.

Appendices B–E; Tables 2–3, pp. 30–32 · Main paper

What to reuse—and what not to infer

Transferable lesson: Teach an abstract method through a solvable special case, then carry the same geometric vocabulary into the general algorithm. Include cases where the simple baseline already works and where the proposed objective has undesirable behavior.

Boundary: Keep exact formulation distinct from numerical enforcement: minibatch Hessian checks, Sinkhorn approximation and unconstrained MLP parameterization do not establish global exactness of the implemented algorithm. Appendix E explicitly discusses this gap.

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

PDF pp. 1–11, Sections 1–6 and reproducibility statement; references pp. 11–13 inspected

Appendix / supplement

Appendices A–F, pp. 14–32, including extra experiments, geometric background, baseline, proofs, implementation and regularization/scaling

Visually inspected PDF pages

3, 8

• Official ICLR 2026 proceedings conference PDF; PDF creation metadata: Mon May 11 11:39:04 2026 EDT · 32 pages