Writing study: Geometry-Aware Similarity Metrics for Neural Representations on Riemannian and Statistical Manifolds
← NeurIPS 2026 oral writing corpus
N Alex Cayco Gajic, Arthur Pellegrino · Official NeurIPS 2026 oral record
Figure, table and section ledger
Editorial contribution type: Empirical analysis, diagnosis & auditing (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
- 13 physical pages
- Before references / appendix
- ≈ 8.13 page-equivalents
- Main figures / tables
- 8 / 0
- Attached appendix figures / tables
- 0 / 0
- Attached appendix span
- ≈ 2.92 page-equivalents
Source-bound correction: Top-level heading and reference/appendix boundaries checked against source PDF text. Bibliography titles/author initials, numbered contribution/property bullets, diagram steps, equations and TOC entries are not substantive headings. Section spans are page-coordinate estimates, not ink area or prose-only length. This is an author manuscript mirror, not a verified accepted-version template. Source layout uses two-column reading order; appendix pp.11–13 is single-column.
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 | ≈ 1.81 pp. | 13.92% |
| 2 Methods | 3 | ≈ 1.73 pp. | 13.31% |
| 3 Numerical experiments | 4 | ≈ 3.37 pp. | 25.92% |
| 4 Mathematical properties | 8 | ≈ 0.41 pp. | 3.15% |
| 5 Discussion | 8 | ≈ 0.5 pp. | 3.85% |
| Acknowledgments | 9 | ≈ 0.09 pp. | 0.69% |
| References | 9 | ≈ 1.95 pp. | 15% |
| Appendix | 11 | ≈ 2.92 pp. | 22.46% |
All detected figure and table locations
Main text
Attached appendix
No recognized caption labels.
Argument and organization
- 1 Introduction and Related work
Motivates intrinsic geometry versus extrinsic point-cloud similarity.
- 2 Methods
Defines pullback geometry, spectral ratio and integrated MSA.
- 3 Numerical experiments
Tests learning regimes, dynamical models and diffusion representations.
- 4 Mathematical properties
States rotation and coordinate invariance.
- 5 Discussion
Limits inference by manifold access, downstream decoding and correlation.
Problem, claim and contribution
The paper motivates an intrinsic-versus-extrinsic geometry distinction rather than claiming that one similarity score is universally superior. It first explains why apparently similar point clouds can hide different computations.
PDF pp.1–2, Figure 1 and contribution framing · Main paper
Argument and section order
Methods move from network-as-manifold map to pullback metric, spectral ratio, and integrated MSA. Three applications then span rich/lazy learning, dynamical models, and diffusion, followed by invariance statements and explicit scope limits.
PDF pp.3–8, Sections 2–5 · Main paper
Figures, captions and visual explanation
Figure 5 aligns the task, extrinsic embedding, intrinsic grid, and numerical similarity in four panels. Figure 8 similarly joins the diffusion schematic, embedding interpolation, image samples, and similarity diagnostics, so readers can relate the scalar measurement to the modeled object.
PDF p.5, Figure 5; p.7, Figure 8 · Main paper
Tables, comparisons and evidence
Hierarchically varying task/regime/seed and trained/untrained dynamics provides interpretable controls. The discussion limits conclusions to known or estimable input manifolds, acknowledges that a hidden-layer geometry ignores downstream use, and calls the analysis correlational rather than causal.
PDF pp.5–6, Figures 6–7; p.8, Discussion · Main paper
Appendix and reproducibility
Appendix A1 provides separately identified proofs for the spectral-ratio pseudodistance, integrated MSA, rotations, and coordinate changes. Main propositions link directly to those arguments rather than relegating the definitions themselves to supplementary material.
PDF pp.11–13, Appendix A1; Propositions 2.2, 2.4, 4.1 and 4.2 · Main paper
Clarity and writing conventions
The method explains a pullback using tangent vectors before its Jacobian expression and distinguishes a Riemannian metric from a distance function. The notation carries through visualizations and invariance statements, supporting readers without assuming the geometric vocabulary is familiar.
PDF pp.3–4, Methods and definitions · Main paper
What to reuse—and what not to infer
Transferable lesson: For a new analysis metric, align a worked geometric example with controlled model comparisons and state which interpretations remain correlational.
Boundary: The source uses a two-column Preprint layout and is not a NeurIPS formatting example. Claims depend on manifold characterization, and a similarity score alone is not causal mechanism evidence. The mirror revision is not confirmed identical to accepted OpenReview.
Apply this lesson using the argument and evidence worksheet.
Inspection record and sources
- Checked
- 2026-10-02
- Reading status
- Writing-focused manuscript scan + selected visuals inspected
- Main text
- Writing-oriented source scan of framing, argument roadmap, selected substantive passages, method/theorem statements, main figure/table captions, evidence and limits across PDF pp.1–8. Two selected main-body pages were actually inspected as rendered images. This is not exhaustive line-by-line technical verification or experiment reproduction. Title and all author names compared against official program metadata and manuscript first page; matching authors, with capitalization, middle initials or expanded given names where applicable.
- Appendix / supplement
- Appendix A1 proof sequence and definitions were read for argument organization; mathematical correctness was not independently established.
- Visually inspected PDF pages
- 5, 7
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.