Writing study: Probabilistic Kernel Function for Fast Angle Testing

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← ICLR 2026 oral writing corpus

Kejing Lu, Chuan Xiao, Yoshiharu Ishikawa · 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) · Theory & formal foundations.

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

≈ 9.49 page-equivalents

Main figures / tables

2 / 1

Attached appendix figures / tables

7 / 4

Attached appendix span

≈ 10.89 page-equivalents

Before references / appendix: ≈ 41.26%References / intervening material: ≈ 11.39%Attached appendix: ≈ 47.35%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	≈ 1.72 pp.	7.48%
2 RELATED WORK	3	≈ 0.39 pp.	1.7%
3 TWO PROBABILISTIC KERNEL FUNCTIONS	3	≈ 0.73 pp.	3.17%
4 ANALYSIS OF PROBABILITY GUARANTEES	4	≈ 1.09 pp.	4.74%
5 IMPLEMENTATION AND COMPLEXITY ANALYSIS	5	≈ 1.7 pp.	7.39%
6 APPLICATIONS TO SIMILARITY SEARCH	7	≈ 1.25 pp.	5.43%
7 EXPERIMENTS	8	≈ 1.37 pp.	5.96%
8 CONCLUSIONS	9	≈ 0.64 pp.	2.78%
ACKNOWLEDGEMENTS	10	≈ 0.08 pp.	0.35%
REFERENCES	10	≈ 2.62 pp.	11.39%
A NOTATIONS	13	≈ 0.39 pp.	1.7%
B PROOF OF LEMMAS	13	≈ 3.99 pp.	17.35%
C SUPPLEMENTARY MATERIALS	17	≈ 2.27 pp.	9.87%
D ADDITIONAL EXPERIMENTS	19	≈ 4.24 pp.	18.43%

All detected figure and table locations

Main text

• Table 1 · p. 8
• Figure 1 · p. 9
• Figure 2 · p. 10

Attached appendix

• Table 2 · p. 13
• Figure 3 · p. 17
• Figure 4 · p. 17
• Table 3 · p. 20
• Table 4 · p. 21
• Table 5 · p. 21
• Figure 5 · p. 21
• Figure 6 · p. 22
• Figure 7 · p. 22
• Figure 8 · p. 23
• Figure 9 · p. 23

Argument and organization

1. Two formal problems

   Separate probabilistic comparison from thresholding before proposing kernels.

2. Kernels, guarantees and implementation

   Connect each definition to a lemma and a practical projection configuration.

3. Applications and experiments

   Evaluate small recall gains separately from graph-search throughput gains.

Problem, claim and contribution

Numbered Problem 1.1 and Problem 1.2 state the decision and computational budget explicitly, preventing angle estimation and angle testing from being conflated.

Section 1, pp. 1–2 · Main paper

Argument and section order

The paper uses a definition–lemma–implementation–application chain, with paired kernels retaining their identities throughout.

Sections 3–6, pp. 3–7 · Main paper

Figures, captions and visual explanation

Appendix Figure 3 translates the routing inequality into lookup columns, arithmetic and a pruning decision. Figure 4 compares three projection geometries using the same query/data markers.

Figures 3–4, p. 17 · Main paper

Tables, comparisons and evidence

Table 1 groups datasets and bolds each best probing result; its caption specifies projection count, top-five probing and ten-run averaging, making small improvements interpretable.

Table 1, p. 8 · Main paper

Appendix and reproducibility

A notation table precedes proofs, then supplementary pseudocode and parameter settings; extra k values and a second graph distinguish robustness checks from the main benchmark.

Appendices A–D, pp. 13–23 · Main paper

What to reuse—and what not to infer

Transferable lesson: When introducing multiple related operators, preserve a one-to-one mapping between problem, definition, theorem, application and experiment. Put comparison protocol in the table caption, especially when gains are small.

Boundary: Do not inherit global speedup language without the operating point: ScaNN wins some high-recall settings, and preprocessing/index overhead is separate from per-candidate cost. The useful method schematics are delayed to the appendix, increasing the main text's entry cost.

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

PDF pp. 1–10, Sections 1–8; references pp. 10–12 inspected

Appendix / supplement

Appendices A–D and LLM-use statement, pp. 13–23: notation, all proofs, supplementary algorithms and all additional experiments

Visually inspected PDF pages

17, 8

• Official ICLR 2026 proceedings conference PDF; PDF creation metadata: Sun Mar 1 16:14:08 2026 EST · 23 pages