Writing study: Foundations of Equivariant Deep Learning: Unifying Graph and Sheaf Neural Networks
← ICML 2026 oral writing corpus
Yoshihiro Maruyama · Official ICML 2026 oral record
Figure, table and section ledger
Editorial contribution type: Theory & formal analysis (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
- 23 physical pages
- Before references / appendix
- ≈ 9.1 page-equivalents
- Main figures / tables
- 0 / 0
- Attached appendix figures / tables
- 0 / 2
- Attached appendix span
- ≈ 12.65 page-equivalents
Source-bound correction: Source-checked physical-page text restores wrapped top-level heading labels on pp. 11. Heading starts and measured positions are unchanged; this is label completeness, not a new section. Newly detected unnumbered impact/singular-acknowledgment statements are retained from the size-aware source extraction; authored heading corrections are unchanged.
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 | ≈ 1.43 pp. | 6.22% |
| 2. Order-Equivariant Neural Networks | 2 | ≈ 3.1 pp. | 13.48% |
| 3. Equivariant Universal Approximation | 5 | ≈ 1.42 pp. | 6.17% |
| 4. Examples: Graphs and Sheaves | 7 | ≈ 2.24 pp. | 9.74% |
| 5. Conclusion | 9 | ≈ 0.47 pp. | 2.04% |
| Impact Statement | 10 | ≈ 0.05 pp. | 0.22% |
| References | 10 | ≈ 1.25 pp. | 5.43% |
| A. Category-Equivariant Neural Networks and Their Relation with OENNs | 11 | ≈ 8.69 pp. | 37.78% |
| B. Proofs | 20 | ≈ 3.96 pp. | 17.22% |
All detected figure and table locations
Main text
No recognized caption labels.
Attached appendix
Argument and organization
- Problem and contribution
Places graph and sheaf constructions in a common order-equivariant setting while separating full, source-labeled pair-state and ordinary anonymous message-passing classes.
- Argument sequence
The linear transporter law is extended through equivariant nonlinear primitives before universality; familiar fixed-graph and sheaf cases then explain the abstraction.
- Evidence and verification
Universality is not transferred to all local message passing: persistent source labels and sufficient communication depth distinguish the positive local result.
Problem, claim and contribution
Places graph and sheaf constructions in a common order-equivariant setting while separating full, source-labeled pair-state and ordinary anonymous message-passing classes.
Contributions, p. 2; Proposition 2.16, p. 5 · Main paper
Argument and section order
The linear transporter law is extended through equivariant nonlinear primitives before universality; familiar fixed-graph and sheaf cases then explain the abstraction.
Sections 2-4, pp. 2-9 · Main paper
Figures, captions and visual explanation
There is no main empirical figure-led narrative: hierarchy inclusions and numbered definitions organize the dense theory, while explicit cyclic/dihedral and vertex-edge equations supply concrete special cases.
Propositions 2.13-2.16, p. 5; Section 4.1-4.2, p. 8; both PNGs inspected · Main paper
Tables, comparisons and evidence
Universality is not transferred to all local message passing: persistent source labels and sufficient communication depth distinguish the positive local result.
Sections 3.2-3.3, pp. 6-7; proof, p. 23 · Main paper
Appendix and reproducibility
The correspondence uses action groupoids, not the thin poset category; a separate activity-recognition example documents subject splits and the relative-rotation information used by its architecture.
Appendix A.2-A.3, pp. 15-19 · Main paper
Clarity and writing conventions
The text repeatedly marks sufficient versus necessary conditions and further-tied special cases, helping prevent a broad unification claim from implying identical expressivity.
Section 4.2, p. 8; Appendix A.2, pp. 15-18 · Main paper
What to reuse—and what not to infer
Transferable lesson: Give a hierarchy of model classes before a universality theorem, then show exactly which extra state or communication assumption enables the stronger result.
Boundary: Approximation is on compact invariant sets for the stated finite group/poset setting; the appendix's category-network illustration is not evidence that every OENN realization learns efficiently or receives equivalent input information.
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 1-5, pp. 1-9: order-equivariant bundle definitions, orbital/pointwise/pair layers, architecture hierarchy, universal approximation and graph/sheaf special cases.
- Appendix / supplement
- Writing-focused appendix scan; code, data and external assets not executed. Appendices A-B, pp. 11-23: category-equivariant framework, precise action-groupoid correspondence, cited activity-recognition illustration and theorem-specific proofs.
- Visually inspected PDF pages
- 5, 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.