Writing study: Characterizing the Discrete Geometry of ReLU Networks

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← ICLR 2026 oral writing corpus

Blake Gaines, Jinbo Bi · 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).

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

27 physical pages

Before references / appendix

≈ 10.11 page-equivalents

Main figures / tables

7 / 1

Attached appendix figures / tables

7 / 2

Attached appendix span

≈ 11.89 page-equivalents

Before references / appendix: ≈ 37.44%References / intervening material: ≈ 18.52%Attached appendix: ≈ 44.04%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	≈ 1.52 pp.	5.63%
2 PRELIMINARIES	3	≈ 1.47 pp.	5.44%
3 THEORETICAL RESULTS ON NETWORK GEOMETRY	4	≈ 2.53 pp.	9.37%
4 ALGORITHM FOR CALCULATING POLYHEDRON BOUNDARIES	7	≈ 1.29 pp.	4.78%
5 EXPERIMENTS	8	≈ 2.18 pp.	8.07%
6 DISCUSSION AND FUTURE WORK	10	≈ 0.52 pp.	1.93%
REFERENCES	11	≈ 5 pp.	18.52%
APPENDIX	16	≈ 0.04 pp.	0.15%
A DETAILED RELATED WORK	16	≈ 1.01 pp.	3.74%
B PROOFS OF THEORETICAL RESULTS	17	≈ 4.25 pp.	15.74%
C ADDITIONAL CATEGORIZATION EXAMPLES	21	≈ 0.33 pp.	1.22%
D ALGORITHMIC DETAILS	21	≈ 0.95 pp.	3.52%
E ADDING BENT HYPERPLANES TO THE DUAL GRAPH	22	≈ 0.71 pp.	2.63%
F EXPERIMENTAL DETAILS	23	≈ 0.81 pp.	3%
G EXTENDED EXPERIMENTAL RESULTS	24	≈ 3.79 pp.	14.04%

All detected figure and table locations

Main text

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

Attached appendix

• Figure 8 · p. 19
• Figure 9 · p. 21
• Figure 10 · p. 23
• Table 2 · p. 23
• Figure 11 · p. 24
• Figure 12 · p. 25
• Table 3 · p. 26
• Figure 13 · p. 27
• Figure 14 · p. 27

Argument and organization

1. Geometric vocabulary

   Network output, polyhedral regions, connectivity graphs, and degree histograms introduce successive representations of one object.

2. Bounds and construction

   Definitions and theorem sketches lead to an enumeration algorithm, controlled experiments, and qualified discussion.

Problem, claim and contribution

The gap is not how many linear regions exist but how they connect, sharply separating this contribution from region-counting literature.

Pages 1–2, introduction. · Main paper

Argument and section order

Parallel theoretical-properties and empirical-observations lists explicitly separate proved statements from patterns found in experiments.

Page 2, contribution lists. · Main paper

Figures, captions and visual explanation

Four matched panels translate one small network from surface to partition to graph to histogram; repeated region letters carry the reader between representations.

Page 2, Figure 1. · Main paper

Tables, comparisons and evidence

The plots place theoretical bounds alongside measured connectivity and preserve dimension colours across panels, making empirical proximity to a bound visible.

Page 8, Figures 4–5. · Main paper

Appendix and reproducibility

The supplement progresses from formal assumptions and proofs to explanatory graph cartoons, LP implementation, training settings, and expanded distributions.

Pages 17–27, appendices B–G. · Main paper

Clarity and writing conventions

Annotated braces explain matrix-expression factors before pseudocode uses them, reducing the notation burden.

Page 7, equations 2–3 and Algorithm 1. · Main paper

What to reuse—and what not to infer

Transferable lesson: For theoretical papers, introduce one concrete object through multiple linked views and visibly distinguish theorems from empirical observations.

Boundary: The results assume genericity/supertransversality and fully connected ReLU structure; they are not immediately transformer or nonlinear-activation guarantees. Real-data enumeration is partial for two datasets, and graph diameter is estimated rather than always computed exactly.

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 1–6; references pages 11–15.

Appendix / supplement

Pages 16–27, appendices A–G: related work, all proofs, categorization examples, LP details, graph updates, training settings, extended results.

Visually inspected PDF pages

2, 8

• ICLR 2026 final proceedings · 27 pages