Writing study: A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization

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← ICLR 2026 oral writing corpus

Hideaki Kim, Tomoharu Iwata · 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) · Inference, algorithms & systems efficiency.

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

24 physical pages

Before references / appendix

≈ 10.23 page-equivalents

Main figures / tables

0 / 4

Attached appendix figures / tables

3 / 4

Attached appendix span

≈ 10.89 page-equivalents

Before references / appendix: ≈ 42.63%References / intervening material: ≈ 12%Attached appendix: ≈ 45.38%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	≈ 0.99 pp.	4.13%
2 PROPOSED METHOD	2	≈ 3.73 pp.	15.54%
3 RELATED WORK	6	≈ 1.06 pp.	4.42%
4 EXPERIMENTS	7	≈ 3.11 pp.	12.96%
5 CONCLUSION	10	≈ 0.63 pp.	2.63%
REPRODUCIBILITY STATEMENT	11	≈ 0.13 pp.	0.54%
REFERENCES	11	≈ 2.87 pp.	11.96%
A EXAMPLES OF THE ESTIMATED TRIGGERING KERNELS ON SYNTHETIC	14	≈ 1 pp.	4.17%
B PROOF OF THEOREM 1 VIA MERCER’S THEOREM	15	≈ 1.14 pp.	4.75%
C PROOF OF THEOREM 1 VIA PATH INTEGRAL REPRESENTATION	16	≈ 1.07 pp.	4.46%
D PROOF OF PROPOSITION 2	17	≈ 1.44 pp.	6%
E PROOF OF PROPOSITION 3	18	≈ 1.06 pp.	4.42%
F PROOF OF PROPOSITION 4	19	≈ 1.42 pp.	5.92%
G EXISTENCE OF EQUIVALENT KERNELS	21	≈ 0.51 pp.	2.13%
H ADDITIONAL EXPERIMENTS	21	≈ 2.03 pp.	8.46%
I DEEP NEURAL NETWORK MODELS VS. KERNEL METHODS	23	≈ 1.22 pp.	5.08%

All detected figure and table locations

Main text

• Table 1 · p. 8
• Table 2 · p. 9
• Table 3 · p. 9
• Table 4 · p. 10

Attached appendix

• Figure 1 · p. 14
• Figure 2 · p. 14
• Table 5 · p. 21
• Table 6 · p. 22
• Table 7 · p. 22
• Figure 3 · p. 22
• Table 8 · p. 23

Argument and organization

1. Representer theorem, constructive approximation, and complexity

   Connects an infinite-dimensional estimator to equivalent kernels, finite features, and an executable linear solve.

2. Controlled processes and real event data

   Separates kernel recovery, predictive likelihood, and computational efficiency across different regimes.

Problem, claim and contribution

The introduction explains why integrated, history-dependent point-process losses break the classical representer argument before specifying the linear least-squares regime that restores tractability.

Sections 1–2.2 · Main paper

Argument and section order

The theorem, baseline-intensity proposition, finite-feature construction, and final estimator form a useful dependency chain. Short main-text proof sketches link to full derivations rather than interrupting that chain.

Theorem 1 and Propositions 2–4, pp. 3–5 · Main paper

Figures, captions and visual explanation

The appendix uses matching 3×3 grids for the interaction kernels, with dashed ground truth and consistent method colors. This exposes where a global error score conceals failures to recover inhibitory or delayed effects.

Figures 1–2, p. 14 · Main paper

Tables, comparisons and evidence

Tables pair recovery error with CPU seconds, disclose trial counts and standard errors, and distinguish statistically non-different results from simply highlighting the numerically best entry.

Tables 1–4, pp. 8–10 · Main paper

Appendix and reproducibility

Two proof routes serve different readers; separate scalability tests vary event count and process dimension, making the cubic dimensional cost visible rather than hiding it behind event-count gains.

Appendices B–G and H.1–H.2 · Main paper

Clarity and writing conventions

The conclusion candidly states that nonnegative intensity requires post-hoc clipping and that learning guarantees remain open. The experiments also disclose excluded unstable baseline trials, which matters for comparison.

Section 4.1.2 and limitations, pp. 9–10 · Main paper

What to reuse—and what not to infer

Transferable lesson: Organize a theory-to-algorithm paper as a dependency chain, and distinguish approximation cost, statistical error, and different scaling dimensions in both prose and experiments.

Boundary: Unity dual coefficients do not make fitting data-independent: the equivalent kernels depend on the observations and require a matrix solve. Exact least-squares theory is implemented with finite random features; clipped intensities and excluded baseline outliers require explicit qualification.

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. 1–11, Sections 1–5, limitations and reproducibility statement.

Appendix / supplement

PDF pp. 14–24, Appendices A–I: fitted kernels, both theorem derivations, all proposition proofs, existence, scalability, and neural-model comparison.

Visually inspected PDF pages

3, 8, 14

• ICLR 2026 official proceedings PDF, retrieved 2026-09-19 · 24 pages