Writing study: Riemannian Metric Matching for Scalable Geometric Modeling of Distributions

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← ICML 2026 oral writing corpus

Jacob Bamberger, Adam Gosztolai, Pierre Vandergheynst, Michael M. Bronstein, Iolo Jones · Official ICML 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: Training, objectives & optimization (primary).

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

20 physical pages

Before references / appendix

≈ 9.17 page-equivalents

Main figures / tables

5 / 1

Attached appendix figures / tables

5 / 0

Attached appendix span

≈ 8.91 page-equivalents

Before references / appendix: ≈ 45.85%References / intervening material: ≈ 9.6%Attached appendix: ≈ 44.55%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	≈ 1.58 pp.	7.9%
2. Background	2	≈ 0.63 pp.	3.15%
3. Conditional metric matching	3	≈ 1.59 pp.	7.95%
4. Convergence and the Riemannian toolkit	5	≈ 1.29 pp.	6.45%
5. Experiments	6	≈ 2.33 pp.	11.65%
6. Related work	8	≈ 1 pp.	5%
7. Conclusion	9	≈ 0.29 pp.	1.45%
Acknowledgements	10	≈ 0.08 pp.	0.4%
Impact statement	10	≈ 0.05 pp.	0.25%
References	10	≈ 1.91 pp.	9.55%
A. Proofs	12	≈ 2.38 pp.	11.9%
B. Extended comparison to higher order score matching.	14	≈ 0.62 pp.	3.1%
C. Low rank loss derivation	15	≈ 0.57 pp.	2.85%
D. Training Algorithms	15	≈ 0.69 pp.	3.45%
E. Experimental details	16	≈ 1.95 pp.	9.75%
F. Additional results	18	≈ 2.7 pp.	13.5%

All detected figure and table locations

Main text

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

Attached appendix

• Figure 5 · p. 17
• Figure 6 · p. 18
• Figure 7 · p. 19
• Figure 8 · p. 19
• Figure 9 · p. 20

Argument and organization

1. Problem and contribution

   Distinguishes learning geometry directly from extracting it through neighborhood graphs or expensive Jacobians of pretrained models.

2. Argument sequence

   Geometric definitions lead to a tractable conditional loss, then convergence and factorization guarantees, followed by dimension, tangent and interpolation tests.

3. Evidence and verification

   The local-manifold/smooth-density theorem and low-rank condition are stated explicitly; the runtime table names one H100 and milliseconds instead of generalizing hardware-independent speed.

Problem, claim and contribution

Distinguishes learning geometry directly from extracting it through neighborhood graphs or expensive Jacobians of pretrained models.

Introduction and contribution list, p. 2 · Main paper

Argument and section order

Geometric definitions lead to a tractable conditional loss, then convergence and factorization guarantees, followed by dimension, tangent and interpolation tests.

Sections 2-5, pp. 2-8 · Main paper

Figures, captions and visual explanation

The dense derivation page gives equations stable numbers; the results page aligns four dataset panels and reports time/memory separately, retaining out-of-memory entries.

Equations 1-6, p. 3; Figure 4 and Table 1, p. 8; both PNGs inspected · Main paper

Tables, comparisons and evidence

The local-manifold/smooth-density theorem and low-rank condition are stated explicitly; the runtime table names one H100 and milliseconds instead of generalizing hardware-independent speed.

Theorems 4.1 and 4.4, p. 5; Table 1, p. 8 · Main paper

Appendix and reproducibility

A dedicated comparison derives the relationship to higher-order score matching; later interpolation panels distinguish latent dimensions and VAE parameter sizes.

Appendix B, p. 14; Figure 9, p. 20 · Main paper

Clarity and writing conventions

The text separates finite-data kernel estimates from limiting geometric recovery, but visual interpolation plausibility is sometimes phrased as global consistency.

Introduction, p. 2; Section 5.3, p. 8 · Main paper

What to reuse—and what not to infer

Transferable lesson: Give a learned estimator its mathematical target, limiting assumptions and measured computational contract before interpreting downstream visual examples.

Boundary: A small-bandwidth theorem at an optimized loss is not a finite-trained-network guarantee; smooth MNIST interpolation does not prove globally correct geometry.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-10-07

Reading status

Writing-focused manuscript scan + selected visuals inspected

Main text

Writing-focused distributed manuscript scan, not exhaustive technical/proof validation. Sections 1-7, pp. 1-9: geometric target, conditional objective, convergence/low-rank guarantees, empirical geometry, costs and related work.

Appendix / supplement

Writing-focused appendix scan; code, data and external assets not executed. Appendices A-F, pp. 12-20: proofs, score-matching relationship, low-rank loss, algorithms, experimental details and additional interpolation results.

Visually inspected PDF pages

3, 8

This is a writing-focused scan of the manuscript and the stated supplementary scope: framing, organization, visual explanation, evidence presentation and wording. It is not a line-by-line proof audit, reproduction of results or exhaustive scientific review.

• Published ICML 2026 proceedings PDF, PMLR volume 306; retrieved 2026-10-07. Public OpenReview decision notes were not independently reverified. · 20 pages