Writing study: FALCON: Few-step Accurate Likelihoods for Continuous Flows

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← ICLR 2026 oral writing corpus

Danyal Rehman, Tara Akhound-Sadegh, Artem Gazizov, Yoshua Bengio, Alexander Tong · 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: Learning & optimization algorithms (primary) · Architecture & representation design · 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

25 physical pages

Before references / appendix

≈ 11.11 page-equivalents

Main figures / tables

7 / 5

Attached appendix figures / tables

10 / 1

Attached appendix span

≈ 9.89 page-equivalents

Before references / appendix: ≈ 44.44%References / intervening material: ≈ 16%Attached appendix: ≈ 39.56%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	≈ 1.57 pp.	6.28%
2 BACKGROUND AND PRELIMINARIES	3	≈ 1.33 pp.	5.32%
3 FEW-STEP ACCURATE LIKELIHOODS FOR CONTINUOUS FLOWS	4	≈ 1.95 pp.	7.8%
4 EXPERIMENTS	6	≈ 3.4 pp.	13.6%
5 RELATED WORK	9	≈ 0.56 pp.	2.24%
6 CONCLUSION	10	≈ 0.77 pp.	3.08%
ACKNOWLEDGEMENTS	11	≈ 0.14 pp.	0.56%
ETHICS STATEMENT	11	≈ 0.11 pp.	0.44%
REPRODUCIBILITY STATEMENT	11	≈ 0.76 pp.	3.04%
REFERENCES	12	≈ 4 pp.	16%
APPENDIX	16	≈ 0.03 pp.	0.12%
A THEORY	16	≈ 1.13 pp.	4.52%
B OTHER FORMULATIONS FOR LAVG	17	≈ 0.36 pp.	1.44%
C EXTENDED RELATED WORK	17	≈ 0.68 pp.	2.72%
D EXPERIMENTAL DETAILS	18	≈ 3.84 pp.	15.36%
E METRICS	22	≈ 0.6 pp.	2.4%
F ADDITIONAL EXPERIMENTS	22	≈ 3.24 pp.	12.96%

All detected figure and table locations

Main text

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

Attached appendix

• Table 6 · p. 18
• Figure 8 · p. 21
• Figure 9 · p. 21
• Figure 10 · p. 21
• Figure 11 · p. 21
• Figure 12 · p. 23
• Figure 13 · p. 24
• Figure 14 · p. 24
• Figure 15 · p. 25
• Figure 16 · p. 25
• Figure 17 · p. 25

Argument and organization

1. Explain why likelihood accuracy changes the task

   Introduces biased molecular samples and importance reweighting before comparing the computational properties of candidate flow families.

2. Separate the mathematical target from the finite model

   Presents flow-map regression, a cycle-consistency regularizer, idealized propositions and a training algorithm, then tests quality, cost and invertibility-related behavior separately.

3. Make cost and quality auditable

   Pairs quality tables with timing and mechanism ablations; the appendix supplies inference pseudocode, metric definitions and the protocol behind each plotted comparison.

Problem, claim and contribution

The paper motivates a specific bottleneck: a molecular proposal must have sufficiently accurate likelihoods for importance reweighting, not merely produce plausible samples. Its contribution bullets separate the objective, architecture, evaluation cost and sample quality. Table 1 compares model families against explicit requirements, making the gap more concrete than a generic faster-generator claim.

Sections 1–3; Table 1 · Main paper

Argument and section order

Preliminaries establish the probability and flow-map notation before the proposed objective. Main-text propositions state the ideal mathematical target, Algorithm 1 describes training, and Appendix D supplies Algorithm 2 for likelihood-aware sampling. The experiments then divide scalability, sample quality and ablations. This train/inference separation is especially useful when the claimed deployment benefit depends on a computation absent from the training diagram.

Sections 2–4; Algorithms 1–2; Appendix D.2 · Main paper

Figures, captions and visual explanation

Figure 1 visually distinguishes a biased proposal from the distribution after self-normalized importance weighting; its caption qualifies equality as an infinite-sample limit. Figure 2 places quality against logarithmic evaluation time rather than reporting an isolated speed ratio. Figure 3 overlays true, unweighted and reweighted energy distributions using consistent colors across four molecules. These are three different evidential jobs: mechanism, cost–quality frontier and distributional diagnosis.

Figures 1–2, PDF page 2; Figure 3, PDF page 7 · Main paper

Tables, comparisons and evidence

Tables 2–3 expose trade-offs that a blanket best-model statement would hide: on alanine dipeptide, ECNF++ has higher effective sample size than FALCON, while FALCON has lower energy Wasserstein distance; FALCON-A has the best displayed torus distance. Larger-molecule comparisons support a different scalability claim. Reported three-seed uncertainty, metric-direction arrows and unavailable entries should survive any summary. Figure 2 concerns evaluation cost; Table 4 includes training plus inference and is not the same speedup.

Section 4.2; Tables 2–4; Figures 2–4 · Main paper

Appendix and reproducibility

Appendix A exposes regularity and exact-objective assumptions; D records architecture, schedules, data splits and inference; E defines periodic and energy-space metrics; F explains each additional experiment. Figure 12 shows an auxiliary inverse model’s reconstruction-learning curve, including raw and smoothed traces. That finite-sample diagnostic is useful evidence, but the heading Proof of Invertibility and the accompanying wording are stronger than a small reconstruction error can establish for the entire learned map.

Appendices A, D–F; Figure 12, PDF page 23; Section 4.4 · Main paper

Clarity and writing conventions

The limitations explicitly acknowledge that the trained model does not inherit a general guarantee of correct likelihoods and that larger molecules remain difficult. Keep those qualifications close to the headline result. Several protocol descriptions also need reconciliation: the main text and Appendix D.3 name different force-field settings for some datasets, and Appendix F.1 calls the Figure 6 distance an energy metric although the main figure uses a torus metric. Cite the exact experiment rather than silently harmonizing them.

Section 5; Section 4.1 versus Appendix D.3; Figure 6 versus Appendix F.1 · Main paper

What to reuse—and what not to infer

Transferable lesson: Build a method paper around an explicit validity contract: which quantity must be accurate, why it matters downstream, and what the new computation changes. Use separate visuals for the mechanism, the cost–quality frontier and failure-sensitive diagnostics; connect idealized guarantees to their assumptions and distinguish them from finite-model measurements.

Boundary: The experiments concern small peptide systems using 4–16 generation steps, not natural images or WSI. Exact Jacobian calculations for a discrete map do not by themselves prove that an approximately trained map is globally invertible. Idealized propositions and low reconstruction errors are different evidence; evaluation-only and training-plus-inference timings must not be combined into one universal speedup.

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

All Sections 1–6, ethics and reproducibility statements, PDF pages 1–11: likelihood-cost motivation, preliminaries, objectives, propositions, training algorithm, molecular evaluations, ablations and limitations.

Appendix / supplement

All Appendices A–F, PDF pages 16–25: assumptions and proofs, objective variants, related formulations, architecture/data/sampling details, metric definitions and all additional experiments.

Visually inspected PDF pages

2, 7, 23

• Official ICLR 2026 proceedings manuscript including appendices · 25 pages