Writing study: It's All Just Vectorization: einx, a Universal Notation for Tensor Operations

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← ICLR 2026 oral writing corpus

Florian Fervers, Sebastian Bullinger, Christoph Bodensteiner, Michael Arens · 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: Inference, algorithms & systems efficiency (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

≈ 10.11 page-equivalents

Main figures / tables

0 / 2

Attached appendix figures / tables

0 / 2

Attached appendix span

≈ 7.89 page-equivalents

Before references / appendix: ≈ 50.55%References / intervening material: ≈ 10%Attached appendix: ≈ 39.45%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	≈ 1.71 pp.	8.55%
2 RELATED WORKS	3	≈ 1.39 pp.	6.95%
3 VECTORIZATION	4	≈ 1.59 pp.	7.95%
4 EINX	6	≈ 3.25 pp.	16.25%
5 COMPARISON WITH EINSUM AND EINOPS	9	≈ 1.29 pp.	6.45%
6 CONCLUSION	10	≈ 0.29 pp.	1.45%
REFERENCES	11	≈ 2 pp.	10%
APPENDIX	13	≈ 0.24 pp.	1.2%
A COMPARISON WITH OTHER EIN*-NOTATIONS	13	≈ 0.87 pp.	4.35%
B COMPARISON WITH NUMPY-LIKE NOTATION	14	≈ 0.5 pp.	2.5%
C ADDITIONAL EXAMPLES OF EINX OPERATIONS	14	≈ 1.39 pp.	6.95%
D DESCRIPTION OF EINX COMPILER	16	≈ 1 pp.	5%
E ADDITIONAL EXAMPLES OF COMPILED CODE	17	≈ 1.31 pp.	6.55%
F EXAMPLES OF EINX EXCEPTIONS	18	≈ 2.06 pp.	10.3%
G BENCHMARK OF OVERHEAD	20	≈ 0.15 pp.	0.75%
H USAGES STATISTICS OF RELATED LIBRARIES	20	≈ 0.37 pp.	1.85%

All detected figure and table locations

Main text

• Table 1 · p. 2
• Table 2 · p. 3

Attached appendix

• Table 3 · p. 20
• Table 4 · p. 20

Argument and organization

1. Reframe the abstraction

   Uses vectorization to explain why many API operations are one primitive with different axis mappings.

2. Notation, implementation and worked comparisons

   Builds syntax from loops, demonstrates practical benefits, then documents generated code and costs.

Problem, claim and contribution

The opening contrasts a reduction call with explicit loops before naming the abstraction, giving readers a familiar problem rather than beginning with new syntax.

Section 1, p. 1 · Main paper

Argument and section order

Lifting and decomposition are explained in opposite directions before syntax is introduced. The later notation can therefore be interpreted rather than memorized from examples alone.

Sections 3.1–4.1, pp. 4–7 · Main paper

Figures, captions and visual explanation

Code blocks serve as the principal visual device: matched attention implementations use gray panels, colored expressions and corner labels, with identical axis meanings stated once.

Section 5.2, p. 10 · Main paper

Tables, comparisons and evidence

The operation-coverage matrix and concrete silent-failure example support different claims; timing is separately decomposed into first-call compilation and cache lookup.

Table 2, p. 3; Section 4.3, p. 8; Table 3, p. 20 · Main paper

Appendix and reproducibility

The appendix roadmap leads to compiler stages, emitted code and actual error messages. These are unusually actionable implementation documentation, not merely additional benchmark rows.

Appendices D–F, pp. 16–19 · Main paper

Clarity and writing conventions

The historical distinction between Einstein summation and axis-string notation prevents an attractive name from substituting for a precise conceptual lineage.

Section 2.1, p. 3 · Main paper

What to reuse—and what not to infer

Transferable lesson: For a software-method paper, explain one unifying semantic rule, demonstrate equivalent code side by side, and publish representative outputs and failure messages as evidence of usability.

Boundary: Readability and fewer silent errors are illustrated, not established by a controlled user study. The universality claim concerns the defined vectorization model; backend capability and the identity-only concatenation restriction still matter. Three NumPy timing examples do not establish universal speedups.

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 including notation, implementation and attention case study.

Appendix / supplement

Pages 13–20, Appendices A–H including compiler, generated code, errors, overhead and usage statistics.

Visually inspected PDF pages

10, 20

• Official ICLR 2026 proceedings conference PDF; PDF creation metadata: Thu Feb 26 11:34:32 2026 EST · 20 pages