CurveFlow: Curvature-Guided Flow Matching for Image Generation

Drake Du; Hao Huang; Mengyu Wang; Yan Luo; Yi Fang

REVIEW 3 major objections 5 minor 2 cited by

CurveFlow: Curvature-Guided Flow Matching for Image Generation

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Replacing straight-line flow paths with curvature-penalized curved paths improves both image quality and caption following in text-to-image generation.

desk verdict A genuinely new regularizer idea for flow matching, but the paper is unverifiable as submitted: the CurveFlow rows in all four tables are placeholders, and the geometric derivation swaps true curvature for a coefficient determinant without justification. read the letter →

arxiv 2508.15093 v2 pith:Y4HVT2EQ submitted 2025-08-20 cs.CV

Yan Luo , Drake Du , Hao Huang , Yi Fang , Mengyu Wang This is my paper · ORCID

classification cs.CV

keywords CurveFlowcurvature-guidedmatchingtext-to-imagegenerationsemanticalignmentrectifiedtrajectoryregularizationinstructioncomplianceCOCOcaptionmetrics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

CurveFlow argues that the straight-line interpolation at the heart of rectified flow is a geometric handicap for text-to-image generation: a zero-curvature path drags the generative process through low-probability regions and washes out fine semantic details. The paper proposes replacing the line with a curved path x_t = a1(t)x0 + a2(t)epsilon whose scaling functions a1(t) and a2(t) are learned by small MLPs, and trains with a flow-matching loss plus a robust curvature penalty that suppresses rapid turning of the coefficient functions. On MS COCO 2014 and 2017, fine-tuning a rectified flow transformer with this objective yields lower FID and consistently higher caption-consistency scores (BLEU, METEOR, ROUGE, CLAIR) than the linear rectified flow baselines and than the nonlinear Rectified Diffusion baseline. The takeaway the paper defends is that curvature-aware planning of the transport path improves instructional compliance without adding sampling steps or inference cost.

What carries the argument

The load-bearing object is the two-term interpolant x_t = a1(t)x0 + a2(t)epsilon together with the regularizer R = lambda * integral_0^1 |a1'(t)a2''(t) - a2'(t)a1''(t)|^2 dt. The cross term a1'a2'' - a2'a1' is the coefficient-space determinant that would multiply the cross product x0 × epsilon in the sample-space curvature formula; the paper replaces that sample-dependent numerator with the same Wronskian determinant of the learned coefficients, squared and integrated over time. The determinant vanishes when both coefficient functions are linear in t—the rectified-flow straight line—and grows when the relative mixing of data and noise reverses, so penalizing it pushes the learned path toward

What would settle it

Train CurveFlow with a1 and a2 constrained to curves that have substantial geometric curvature but zero Wronskian determinant—for example a1 = a2, so the penalty is zero yet the paths are bent—and compare against the paper's full method. If these zero-penalty curved models still beat rectified flow, the determinant penalty is not the operative mechanism. Alternatively, compute the empirical average of the true curvature in Eq. 4 on held-out pairs and check whether it correlates with CLAIR score across regularizer strengths; a null correlation would falsify the causal story.

Watch

Extended reading notes

Core claim

The central claim is that flow trajectories should be allowed to curve, and that the right kind of curvature guidance is a penalty on the intrinsic turning of the coefficient functions rather than on the noisy per-sample curvature expression. Concretely, the paper constructs paths x_t = a1(t)x0 + a2(t)epsilon and trains with a velocity-matching objective plus a regularization term lambda * integral_0^1 |a1'(t)a2''(t) - a2'(t)a1''(t)|^2 dt. Penalizing the determinant a1'a2'' - a2'a1'', a Wronskian-like measure of how fast the relative contributions of data and noise rotate, stabilizes training and controls path geometry without depending on individual sampled x0 and epsilon. In the reported e

Load-bearing premise

The argument stands or falls on the assumption that penalizing the coefficient determinant a1'(t)a2''(t) - a2'(t)a1''(t) is a faithful proxy for the path curvature that hurts semantic alignment; if that determinant does not track the true trajectory curvature, the claimed mechanism has no support.

Editorial extensions

If this is right

If CurveFlow is right, text-to-image flow matching models can be made more instruction-compliant by training on curved, curvature-penalized paths instead of straight-line or merely relaxed paths, without changing the base model or sampling procedure.
The regularization strength matters: the paper reports monotonic improvements in METEOR and ROUGE-1 up to lambda ≈ 0.05, with relative gains of about 5–6 percent, so trajectory geometry is a tunable degree of freedom in flow matching.
The added cost is confined to training: the coefficient networks and the finite-difference Riemann sum add roughly 8 GPU hours relative to the unrewetted rectified flow baseline, while inference remains identical to standard ODE solving.
Nonlinear trajectories alone are not enough; the comparison against Rectified Diffusion suggests that explicit curvature regularization, not just relaxing straightness, is what delivers the consistent gains across caption-similarity metrics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The regularizer is a surrogate: R penalizes the coefficient-space determinant, whereas the true sample-space curvature of Eq. 4 also involves the noisy cross product and speed. A direct test of the paper's mechanism would check whether this determinant tracks the empirical average of the true curvature across learned trajectories; the paper does not report that correlation.
Because nothing in the loss is specific to images or COCO captions, the same curved-flow-plus-curvature-penalty objective transfers to audio, video, or 3D flow matching. If the curvature–semantics link is real, curved flows should also improve any task where the output must respect structured input constraints.
The design contains a nested simpler hypothesis: much of the gain may come from smoothing a1 and a2 at all, rather than from the Wronskian cross-coupling. Comparing R with a decoupled penalty on a1'' and a2'' separately would reveal whether the determinant form is essential or merely a convenient way to smooth both schedules.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Hey,

First, the one-line verdict: the paper proposes a genuinely new regularizer for flow matching—penalizing the coefficient determinant a1'a2'' - a2'a1'' on the learned interpolation x_t = a1(t)x0 + a2(t)epsilon—but you can't actually check whether it works because the CurveFlow rows in Tables 1-4 are placeholder boxes. The prose reports specific numbers (29.54 BLEU-1, 64.26 CLAIR, FID 20.57 on COCO17) but the tables don't contain them. That's a missing dataset, not a minor presentation problem.

What's worth taking seriously: the idea is clearly distinct from Rectified Diffusion, which relaxes straightness without this particular curvature penalty. The authors also cite the relevant prior work (Esser et al., Wang et al.) and they set up a reasonable evaluation pipeline, even if the proxy—caption generated images with BLIP-2, then match captions to references with BLEU/METEOR/ROUGE/CLAIR—can reward generic captions. The qualitative examples in Figure 3, if representative, show the kind of instruction-following improvement they're after.

Now the soft spots, in order of size. First, the empirical claim is unverifiable as submitted. No code, no seeds, no error bars, and the tables are empty for their own method. Second, the geometry is underdeveloped. Eq. 4 uses a cross product in a high-dimensional latent space, where that formula isn't the standard definition of curvature. Eq. 6 replaces that with a Wronskian-style determinant of the coefficient functions, but there's no proof that this proxy controls the trajectory geometry that matters for semantic alignment. It might be a fine heuristic; it's just not established. Third, the evaluation's indirectness means even the filled-in numbers would need scrutiny.

Who is this for? Someone working on flow trajectory design might find the regularization idea worth a quick reimplementation. But the paper as written doesn't deserve a serious referee — it's incomplete. I'd desk reject or send back for major revision with a demand for the actual tables and a clearer derivation. The underlying idea is coherent enough to retest, but the submission is not there yet.

Let's talk soon.

Referee Report

3 major / 5 minor

Summary. The paper proposes CurveFlow, a flow-matching variant that replaces the straight-line interpolation of rectified flow with a curved trajectory x_t = a1(t)x0 + a2(t)ε, where a1 and a2 are learned coefficient functions. A 'robust curvature regularization' term penalizes the determinant a1'(t)a2''(t) − a2'(t)a1''(t), intended to control trajectory curvature without sample-dependent variance. The method is evaluated on MS COCO 2014 and 2017 for text-to-image generation, using image-to-text metrics (BLEU, METEOR, ROUGE, CLAIR) computed from BLIP-2 captions, plus FID, IS, and CLIPScore. The paper claims state-of-the-art performance over rectified flow variants and Rectified Diffusion, with best FID values of 20.57 (COCO17) and 10.44 (COCO14).

Significance. If the empirical claims were fully supported, curvature-guided flow trajectories would be a simple and potentially useful extension to flow matching, with evidence that non-linear paths improve both image quality and semantic alignment. The paper also ships a public code repository and includes qualitative examples. However, the central experimental evidence is absent from the submitted tables: every CurveFlow row is a placeholder, so none of the headline numbers or rankings can be checked. In addition, the mathematical link between the proposed regularizer and the actual trajectory curvature is not established. The idea is worth considering, but the manuscript in its current form does not demonstrate its central claim.

major comments (3)

[Tables 1-4 and 'Experimental Results'] All CurveFlow rows in Tables 1-4 contain placeholder boxes rather than numeric entries, yet the text reports specific values (e.g., 29.54 BLEU-1 and 64.26 CLAIR on COCO17; FID 20.57 on COCO17 and 10.44 on COCO14). Without numeric table entries, the claims of state-of-the-art performance, superiority over Rectified Diffusion, and best FID are unverifiable. This is the load-bearing evidence of the paper and must be completed with actual numbers and, ideally, error bars or run details.
[Section 'CurveFlow Trajectory', Eq. (4) vs Eq. (6)] Eq. (4) defines curvature using a cross product x0 × ε. In the high-dimensional latent space used for image generation (d > 3), this cross product is not a standard operation; the usual curvature formula for a curve in R^n is κ = sqrt(|γ'|^2|γ''|^2 − (γ'·γ'')^2)/|γ'|^3. Eq. (6) then replaces the curvature with the determinant a1'a2'' − a2'a1'', which is a Wronskian in coefficient space. No proof or argument is given that this Wronskian is a faithful proxy for the geometric curvature of the trajectory in data space. As written, the central mechanism linking the regularization to trajectory geometry is unsupported.
[Section 'Dataset'] The experiments are described as using 'a training set containing approximately 11K images' for COCO, which is far smaller than the standard COCO train split. It is unclear whether this is a deliberate subset, whether all baselines and CurveFlow are trained on exactly the same subset, and how this affects comparability with published results. This should be clarified, since the evaluation protocol affects the fairness of the claimed improvements.

minor comments (5)

[Abstract and code availability] The abstract gives the code URL as github.com/Harvard-AI-and-Robotics-Lab/CurveFlow, while the full text gives github.com/anonymous4science/CurveFlow. This inconsistency is confusing and should be resolved, especially if the submission is under double-blind review.
[Throughout] Several instances of the model name appear as '��' in headings and text, likely a rendering problem. These need to be fixed to read 'CurveFlow'.
[Section 'Preliminary'] There is a duplicated parenthesis in 'Rombach et al. 2022))' and the second derivative expression is garbled: '� � ��/�� '. Please clean up the mathematical notation.
[Section 'CurveFlow Training Objective'] The Riemann-sum approximation of Eq. (6) with numerical central differences for derivatives is described only briefly. The step size N and the choice of central differences should be stated in the main text rather than only in the code repository.
[Figure 4] The x-axis of the left panel is described as logarithmic, but the figure caption and axis labels do not state this. Please add explicit axis labels and units for lambda and hidden dimension.

Circularity Check

0 steps flagged · score 1.0 of 10

No meaningful circularity: the loss in Eq. 7 is not defined in terms of the reported metrics, and the only self-citation is motivational rather than load-bearing. The main defect is unverifiable placeholder table entries, which is an evidence problem, not a circularity problem.

full rationale

Walking the derivation chain: CurveFlow defines x_t = a1(t)x0 + a2(t)epsilon (Eq. 3), fits the velocity field with Eq. 5, and adds the robust curvature penalty Eq. 6 to obtain Eq. 7. The reported metrics (BLEU, METEOR, ROUGE, CLAIR, FID, IS, CLIPScore) are computed post hoc on generated images; none of them appears in Eq. 7, and the coefficient networks a1,a2 are not fit to those metrics. The hyperparameter lambda and MLP dimension are selected on the validation split, which is ordinary model selection rather than a fitted-input-called-prediction reduction. Thus there is no self-definitional or construction-level circularity. The paper does cite Wang et al. 2025, which shares an author (M. Wang), as motivation and as the Rectified Diffusion baseline, but the curvature loss itself is derived in this paper and does not reduce to that citation; the cited work is published and externally falsifiable, so the self-citation is not load-bearing for the derivation. The more serious issue is that Tables 1-4 show placeholder boxes ('?????') for every CurveFlow row while the text asserts specific SOTA numbers; this makes the central empirical claim unverifiable as written, but missing evidence is not circular reasoning. Score 1 reflects the minor self-citation and the unsupported empirical presentation without implying a circular derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central method relies on treating latent-space vectors as if cross-product curvature applies, on a Wronskian proxy without derivation, on caption-matching as a measure of instruction following, and on unspecified boundary handling for the learned coefficient functions.

free parameters (3)

curvature regularization strength lambda = 0.1
Selected by validation FID and METEOR on COCO, as shown in Figure 4; performance is sensitive in the range 0.01 to 0.1.
coefficient network hidden dimension = 64
Selected by validation FID and METEOR; 32 is too small and 128 overfits according to Figure 4.
Riemann grid size N = not stated, example 100
Used to approximate the integral in Eq. 6; the paper gives an example N=100 but does not specify the actual value used in experiments.

assumptions (4)

domain assumption Curvature of a high-dimensional latent trajectory can be measured by the cross-product formula ||x_dot cross x_ddot|| / ||x_dot||^3.
Equation 4 uses a cross product between x0 and epsilon in latent space, but standard cross-product curvature is only defined in R^3; the paper does not justify the generalization to high-dimensional latents.
ad hoc to paper Penalizing the Wronskian determinant a1' a2'' - a2' a1'' is a robust proxy for penalizing trajectory curvature and does not introduce harmful variance.
Equation 6 replaces the full curvature expression with the coefficient determinant; no derivation shows that this proxy preserves the semantic benefit of curvature regularization.
domain assumption BLIP-2 generated captions scored by BLEU, METEOR, ROUGE, and CLAIR against original COCO captions measure instructional compliance.
Section on evaluation metrics uses this proxy; N-gram metrics can reward generic captions and do not directly measure whether the image follows the text prompt.
domain assumption The learned MLPs for a1(t) and a2(t) can represent the boundary conditions a1(0)=1, a1(1)=0, a2(0)=0, a2(1)=1 and remain stable under numerical differentiation.
The paper never specifies how the boundary conditions are enforced or how the numerical second derivatives behave at the endpoints of the interval.

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Cite this review

Pith. "Pith review of CurveFlow: Curvature-Guided Flow Matching for Image Generation." pith.science (2026). https://pith.science/paper/Y4HVT2EQ

@misc{pith2026250815093,
  author       = {Pith},
  title        = {Pith review of: CurveFlow: Curvature-Guided Flow Matching for Image Generation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4HVT2EQ}},
  note         = {Machine review of arXiv:2508.15093}
}

read the original abstract

Existing rectified flow models are based on linear trajectories between data and noise distributions. This linearity enforces zero curvature, which can inadvertently force the image generation process through low-probability regions of the data manifold. A key question remains underexplored: how does the curvature of these trajectories correlate with the semantic alignment between generated images and their corresponding captions, i.e., instructional compliance? To address this, we introduce CurveFlow, a novel flow matching framework designed to learn smooth, non-linear trajectories by directly incorporating curvature guidance into the flow path. Our method features a robust curvature regularization technique that penalizes abrupt changes in the trajectory's intrinsic dynamics.Extensive experiments on MS COCO 2014 and 2017 demonstrate that CurveFlow achieves state-of-the-art performance in text-to-image generation, significantly outperforming both standard rectified flow variants and other non-linear baselines like Rectified Diffusion. The improvements are especially evident in semantic consistency metrics such as BLEU, METEOR, ROUGE, and CLAIR. This confirms that our curvature-aware modeling substantially enhances the model's ability to faithfully follow complex instructions while simultaneously maintaining high image quality. The code is made publicly available at https://github.com/Harvard-AI-and-Robotics-Lab/CurveFlow.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

[4] [4]

S.; and Vanden-Eijnden, E

Albergo, M. S.; and Vanden-Eijnden, E. 2023. Building Normalizing Flows with Stochastic Interpolants. In 11th International Conference on Learning Representations, ICLR 2023

work page 2023

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Banerjee, S.; and Lavie, A. 2005. METEOR: An automatic metric for MT evaluation with improved correlation with human judgments. In Proceedings of the acl workshop on intrinsic and extrinsic evaluation measures for machine translation and/or summarization, 65--72

2005

[6] [6]

Chan, D.; Petryk, S.; Gonzalez, J.; Darrell, T.; and Canny, J. 2023. CLAIR: Evaluating Image Captions with Large Language Models. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, 13638--13646

work page 2023

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Choi, J.; Kim, J.; Shim, S.; Lee, J.; and Yoon, K. 2023. Perception Prioritized Training of Diffusion Models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 4567--4576

work page 2023

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Deng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei-Fei, L. 2009. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, 248--255. Ieee

2009