Writing study: On Minimum Depth and Width of Floating-Point Neural Networks for Representing Floating-Point Functions
← ICML 2026 oral writing corpus
Sejun Park, Yeachan Park, Geonho Hwang · 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
- 19 physical pages
- Before references / appendix
- ≈ 8.17 page-equivalents
- Main figures / tables
- 1 / 1
- Attached appendix figures / tables
- 0 / 1
- Attached appendix span
- ≈ 8.91 page-equivalents
Source-bound correction: Source-checked physical-page text restores wrapped top-level heading labels on pp. 3. 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 | ≈ 1.46 pp. | 7.68% |
| 2. Preliminaries | 2 | ≈ 1.06 pp. | 5.58% |
| 3. Minimum depth and width of FP networks for FP function representation | 3 | ≈ 1.82 pp. | 9.58% |
| 4. Proofs | 5 | ≈ 3.16 pp. | 16.63% |
| 5. Conclusion | 8 | ≈ 0.21 pp. | 1.11% |
| Acknowledgement | 9 | ≈ 0.08 pp. | 0.42% |
| Impact Statement | 9 | ≈ 0.05 pp. | 0.26% |
| References | 9 | ≈ 1.91 pp. | 10.05% |
| A. Values of p and q in Practical Floating-Point Formats | 11 | ≈ 0.2 pp. | 1.05% |
| B. Technical lemmas | 11 | ≈ 4.8 pp. | 25.26% |
| C. Additional proofs | 16 | ≈ 3.91 pp. | 20.58% |
All detected figure and table locations
Main text
Attached appendix
Argument and organization
- Problem and contribution
Explains why finite input/output memorization does not inherit exact-arithmetic depth and width results once rounding can destroy injectivity.
- Argument sequence
A comparison table precedes exact numeric semantics; theorem statements are grouped by depth and width, then constructive upper bounds are proved before lower bounds.
- Evidence and verification
Rounding mode, left-associative accumulation, nonfinite values, activation and layer-count conventions are defined before the results, making the formal computational model inspectable.
Problem, claim and contribution
Explains why finite input/output memorization does not inherit exact-arithmetic depth and width results once rounding can destroy injectivity.
Contributions and Table 1, p. 2; discussion, p. 5 · Main paper
Argument and section order
A comparison table precedes exact numeric semantics; theorem statements are grouped by depth and width, then constructive upper bounds are proved before lower bounds.
Sections 2-4, pp. 2-8 · Main paper
Figures, captions and visual explanation
The opening table includes domain, codomain and theorem references for each bound; the proof page uses explicitly numbered claims to organize the collision/NaN lower-bound argument.
Table 1, p. 2; Claims 16-18, p. 8; both PNGs inspected · Main paper
Tables, comparisons and evidence
Rounding mode, left-associative accumulation, nonfinite values, activation and layer-count conventions are defined before the results, making the formal computational model inspectable.
Sections 2.2-2.4, pp. 2-3 · Main paper
Appendix and reproducibility
The remaining proofs rely on dedicated technical constructions for finite floating-point values; appendix destinations are named when the main text states each omitted theorem.
Section 3.2, pp. 4-5; Appendices B-C, pp. 11-19 · Main paper
Clarity and writing conventions
The paper uses restricted-domain counterexamples and exceptions to qualify the headline minimum-depth claim, while explicitly retaining the four-unit gap in general width bounds.
Table 1, p. 2; Theorems 6-9, p. 5; conclusion, p. 8 · Main paper
What to reuse—and what not to infer
Transferable lesson: Put the computational semantics and domains in the comparison table's contract, and distinguish a tight result from a remaining upper/lower gap.
Boundary: Worst-case representability is not trainability, practical model size or runtime; alternate accumulation orders, mixed precision and fused hardware operations are not automatically covered by the defined arithmetic.
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. 1-8: exact-versus-floating arithmetic comparison, number/operation/network definitions, depth/width results and selected constructive/lower-bound proofs.
- Appendix / supplement
- Writing-focused appendix scan; code, data and external assets not executed. Appendices A-C, pp. 11-19: numerical-format support, technical lemmas and remaining theorem/proof constructions.
- Visually inspected PDF pages
- 2, 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.