We thank the reviewer for their thoughtful feedback. It helped us clarify the connection between Section 4 and earlier theory, refine the gradient variance discussion in Section 3.2, and reframe the Gaussian example. We also added new CelebA experiments that support our theoretical findings, especially Proposition 2.

Addressing weaknesses

“The clarity of the paper writing can be improved.”

We thank the reviewer for this valuable feedback. We will revise the manuscript to improve clarity and better emphasize the central role of gradient variance throughout. Specifically, we will improve the exposition of Proposition 1 and revise lines 177–183 to avoid overloading the narrative with serial (A)/(B) references.

We will strengthen the paper’s narrative thread and relation to gradient variance by clarifying the following schema, both textually and through an improved Figure 1:

(1) Lemma 1 (Gaussian closed-form) → Prop. 1 (grad. variance bounds)
(2) Gaussian experiments
(1)+(2) ⇒ Conclusions (low grad.variance ≠ OT; reflow skips intersection) → (generalizes) → Prop. 2 (finite data).

“...but no new method.”
“Straight paths give faster simulation and lower discretization error.”

We agree that straight-line deterministic interpolants are attractive due to their simulation efficiency. While we do not propose a new method, our contribution lies in showing that this efficiency can obscure critical failure modes: in noiseless settings, models can minimize both loss and gradient variance while still learning non-optimal ($W_2$) couplings.

This challenges key assumptions in Rectified Flows:

1. Gradient variance reflects path intersections [3,4]
2. ReFlow corrects such intersections via iteration [5]
3. Two iterations suffice for convergence [1,6]

In finite-data regimes, these assumptions break down:

• No convergence to straight paths after two steps (Counterexample 1)
• Skipping intersections and vanishing variance where paths cross (Fig. 6)
• Loss and variance minimized via memorization of poor couplings (Prop. 2 + CelebA experiment)

While straight paths are efficient, our results show they may never be reached in practice (Fig. 6), highlighting the need for more robust transport strategies.

“The novelty of utilizing noise is not that significant.”

We agree, and we do not claim that noise addition is novel. Instead, it plays a key role in our analysis by disrupting the conditions under which memorization arises (see Prop. 2, and CelebA experiments). The contrast between deterministic and noisy settings helps reveal structural limitations in the loss landscape of ReFlow.

“beneficial to experiment on more datasets”, “to be examined how much drawback reflected by the synthetic Gaussian transport counter examples will truly appear in general flow matching methods”

To address concerns about generalization beyond synthetic settings, we provide new experiments on the CelebA benchmark. These results directly validate Proposition 2 on high-resolution, real-world data and confirm that the observed memorization dynamics persist in complex visual domains, not just toy examples.

Questions

MLP cannot approximate without error. Is this really a critical issue?

The main purpose of Lemma 1 is to give the closed-form OT vector field and its rotated version, which Proposition 1 builds upon. The extra fact about exact representability is not needed for the proof, but shows that even Gaussians yield functions (e.g., 1/x) hard to approximate with standard networks.

Section 3.2 role?

We agree that Section 3.2 is less central to the main narrative. We will move it to the appendix and revise the surrounding text to maintain focus on the paper’s core contribution: understanding failure modes in vector field learning through gradient variance.

Figure 3 caption

Thank you for catching the error in Figure 3’s caption. We meant to convey that as $\sigma$ increases, the optimal solution shows lower variance than the vector field aligned with the pairing.

New Experiments on CelebA

To empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from [2]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.

To test this, we compare:

• CFM (σ = 0): trained with deterministic interpolants
  $x_t = (1 - t) x_0 + t x_1$

• CFM (σ = 0.05): trained with slightly noisy interpolants
  $x_t = (1 - t) x_0 + t x_1 + \sqrt{t(1 - t)} \, \sigma \epsilon$

The small noise ($\sigma = 0.05$) is not for performance, but to break injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling memorization (Prop. 2).

Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants

Setup: To directly test whether CFM with deterministic interpolants ($\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:

• We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].
• We shuffle the targets $(x_1)$, breaking the optimal structure and producing shuffled deterministic pairings $(x_0, S(x_1))$.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the shuffled training targets to assess memorization.

Since the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.

Interpretation:
These results confirm Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) minimizes loss by memorizing training targets—even when the pairings are arbitrarily shuffled. In contrast, CFM with noise ($\sigma = 0.05$) avoids this and generalizes better to the true OT map.

Extra comment:
Memorization decreases with larger datasets because the model size is fixed, making it harder to memorize all pairings. This does not contradict Proposition 2: the objective still favors memorization when possible—larger datasets limit its feasibility, not its tendency.

---

Simulated 1-ReFlow: CFM Memorizes Its Own Outputs

Setup: To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.

• Initial training: We train a base CFM model ($\sigma = 0$) on random, non-OT pairings.
• Integration: Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.

Interpretation:
The results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\sigma = 0.05$), despite only a small amount of noise, avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.

---

References

[1] Saptarshi Roy, Vansh Bansal, Purnamrita Sarkar, and Alessandro Rinaldo. 2-rectifications are enough for straight flows: A theoretical insight into Wasserstein convergence.

[2] Korotin, Alexander, et al. Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark. NeurIPS 34 (2021): 14593–14605.

[3] Fjelde, T., Mathieu, E., and Dutordoir, V. An Introduction to Flow Matching. MLG Blog, January 2024.

[4] Gagneux, A., Martin, S., Emonet, R., Bertrand, Q., and Massias, M. A Visual Dive into Conditional Flow Matching. ICLR Blogposts, April 2025.

[5] Liu, Xingchao, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow.

[6] Lee, Sangyun, Zinan Lin, and Giulia Fanti. Improving the training of rectified flows. NeurIPS 37 (2024): 63082–63109.

Dear Reviewer,

Thank you for your continued input and thoughtful feedback. We agree that reporting FIDs allows our results to be more comparable / easier to validate.

We just ran a quick evaluation on 10,000 samples, and the values for all models' FIDs were within 30-40, indicating that all models have relatively good image quality. We will add a table with these results in the next version of our manuscript.

We are currently in the process of designing (and running) a FID-based benchmark that is suitable for our memorisation/shuffling experiments as an alternative to the precomputed OT pairings from [2]. Unfortunately, we won't be able to finish this by the end of the review period, which is today AoE. Still, we will add these in the next version of our manuscript.

We also appreciate the feedback on the clarity, and do agree that restructuring is necessary to ensure more cohesion with the results, and a stronger flow/conclusion. In our final response to reviewer vzJd (above) we addressed this more thoroughly, providing more precise conclusions and connective tissue.

Thank you for the constructive and thoughtful review. We appreciate your interest in ReFlow’s behavior under different settings. Your feedback helped us clarify the broader implications of our theoretical findings and inspired new experimental directions.

Addressing Weaknesses & Limitations

"The work needs extended experiments with varying dataset sizes, ...” “How ReFlows would behave on datasets given ...” “Some more (counter)examples on CFN suboptimality while achieving a stationary point on loss”

We now include new experiments on CelebA OT pairs from [1] that directly validate Proposition 2 in both adversarial and ReFlow-style settings, varying dataset size and noise scale. These settings allow us to probe the relationship between vector field quality and interpolation strategy, even at stationary points of the loss curve.

“The work lacks some thorough analysis, especially towards SBMs and optimal transport (OT).”

To address this, we evaluate CFM and SBM-like interpolants across four types of pairing:

1. Random pairings
2. Ground-truth OT pairings
3. Shuffled OT pairings: $(x_0, \mathcal{S} (\text{OT}(x_0)))$ , where $\mathcal S$ is a shuffling function
4. CFM-integrated pairings

We assess how closely learned couplings align with the OT map and the pairings used during training. These experiments isolate whether models genuinely recover OT or merely memorize their training signal. We look forward to your thoughts on these results.

Work indicates concrete limitations but hardly discusses how to fix it.

We believe highlighting ReFlow’s limitations is a necessary first step toward addressing its failure modes. Existing literature largely assumes idealized conditions (e.g., infinite data, perfect discretization), while our work shows these assumptions break down in practice. Our added noise experiments—simple yet theoretically grounded—offer a promising direction for improving generalization and vector field quality. We hope this opens the door for further study.

Questions

Can regularization (e.g., KL, LR scheduling) mitigate suboptimality?

We use dropout and gradient clipping, but not weight decay or KL-based regularization. While such techniques can help smooth solutions, they cannot eliminate the root issue: the learning objective permits zero-loss, zero-gradient-variance solutions that perfectly memorize mismatched pairings (see Proposition 2). Crucially, our findings arise even without strong regularization. The contrast between CFM (noiseless) and CFM (noisy) models—trained identically—highlights how interpolant choice fundamentally affects learning, independent of regularization strategy.

---

New Experiments on CelebA

To empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from Korotin et al. [1]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.

To test this, we compare:

• CFM (σ = 0): trained with deterministic interpolants
  $x_t = (1 - t) x_0 + t x_1$

• CFM (σ = 0.05): trained with slightly noisy interpolants
  $x_t = (1 - t) x_0 + t x_1 + \sqrt{t(1 - t)} \, \sigma \epsilon$

The small amount of noise ($\sigma = 0.05$) is not introduced for performance reasons but to break the injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling the memorization pathway outlined in Proposition 2. Both models use identical U-Net architectures.

---

Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants

Setup: To directly test whether CFM with deterministic interpolants ($\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:

• We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].
• We shuffle the targets $(x_1)$, breaking the optimal structure and producing random pairings $(x_0, S(x_1))$ with no coherent transport semantics.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the shuffled training targets to assess memorization.

Since the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.

Interpretation:
The results directly support Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) minimizes loss by tightly memorizing its training targets, even when they are arbitrarily shuffled and non-optimal. In contrast, CFM with small noise ($\sigma = 0.05$) resists this memorization trap and generalizes significantly better to the true OT targets.

Extra comments:

• Memorization decreases with larger datasets, as expected, since model capacity is fixed, it becomes harder to memorize all pairings in the 50K setting.
• However, this does not contradict Proposition 2: the optimization still favors memorization when possible. Larger datasets simply make perfect memorization less feasible, not less likely as an objective.

---

Simulated 1-ReFlow: CFM Memorizes Its Own Outputs

Setup: To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.

• Initial training: We train a base CFM model ($\sigma = 0$) on random, non-OT pairings.
• Integration: Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.

Interpretation:
The results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\sigma = 0.05$), despite only a small amount of noise, avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.

---

References

[1] Korotin, Alexander, et al. Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark. NeurIPS 34 (2021): 14593–14605.

We thank Reviewer yZa2 for the thoughtful and constructive feedback. We're glad you found the theoretical contributions and gradient variance analysis compelling. Your comments helped us clarify the connection between our results and recent developments, and motivated us to conduct additional experiments on CelebA to broaden the empirical support.

---

Addressing Weaknesses

"is the limited experimental scope"

We now include new experiments on CelebA OT pairs from [1] that directly validate Proposition 2 in both adversarial and ReFlow-style settings (see the end of the rebuttal for details).

"Practical impact feels somewhat incremental."

While adding noise is not novel, our contribution lies in bridging idealized Rectified Flows (infinite data) with practical settings (finite data), where failure modes emerge:

• no convergence to straight paths after two iterations (Counterexample 1),
• skipping intersections at inference (Fig. 6),
• vector fields minimizing both loss and variance reduce to memorization (Prop. 2). The noise addition is not our contribution per se, but rather a principled choice aligned with the theory: Prop. 2 holds only under deterministic (noiseless) interpolants.

The connection between gradient variance and actual transport quality could be made more intuitive.

We will strengthen the paper’s narrative thread by clarifying the following schema, both textually and through an improved Figure 1:

(A) Lemma 1 (Gaussian closed-form) → Prop. 1 (grad. variance bounds)
(B) Gaussian experiments
(A)+(B) ⇒ Conclusions (low grad. ≠ OT; reflow skips intersection) → (generalizes) → Prop. 2 (finite data).

Questions

“Provide additional results…”

We provide additional results on CelebA (higher resolution images) OT pairs from [2]

Gradient Variance as a Tool (increase Practicality)

We agree that utilizing gradient variance as a diagnostic tool could offer a more constructive perspective on the paper. One direction (Fig. 3) is using gradient variance to tune the noise scale during training. For example, one could identify the point at which the gradient variance of a model trained with noise (e.g., SBM) becomes lower—around minima—than that of a model trained with noiseless interpolants. We view this as a separate line of inquiry.

Our motivation for studying gradient variance over time stems from how the term “intersections” has been heavily emphasized in the Rectified Flow literature. In practice, however, intersections almost never occur in high-dimensional spaces with random samples (as formally proven in Proposition 3, Appendix, p. 25). Within this context, gradient variance provides a more meaningful lens for analysis than the geometry of interpolant intersections.

“Comparison, consistency models…”

Although we did not directly evaluate consistency models in this work, prior research suggests they avoid many of the failure modes we observe in Rectified Flows. Consistency models train a one/k-step mapping, avoiding iterative dynamics that can lead to memorization, but even worse, to degradation of the target as more iterations are put in place.

New Experiments on CelebA

To empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from Korotin et al. [1]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.

To test this, we compare:

• CFM (σ = 0): trained with deterministic interpolants
  $x_t = (1 - t) x_0 + t x_1$

• CFM (σ = 0.05): trained with slightly noisy interpolants
  $x_t = (1 - t) x_0 + t x_1 + \sqrt{t(1 - t)} \, \sigma \epsilon$

The small amount of noise ($\sigma = 0.05$) is not introduced for performance reasons but to break the injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling the memorization pathway outlined in Proposition 2. Both models use identical U-Net architectures.

---

Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants

Setup: To directly test whether CFM with deterministic interpolants ($\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:

• We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].
• We shuffle the targets $(x_1)$, breaking the optimal structure and producing random pairings $(x_0, S(x_1))$ with no coherent transport semantics.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the shuffled training targets to assess memorization.

Since the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.

Interpretation:
The results directly support Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) minimizes loss by tightly memorizing its training targets, even when they are arbitrarily shuffled and non-optimal. In contrast, CFM with small noise ($\sigma = 0.05$) resists this memorization trap and generalizes significantly better to the true OT targets.

Extra comments:

• Memorization decreases with larger datasets, as expected, since model capacity is fixed, it becomes harder to memorize all pairings in the 50K setting.
• However, this does not contradict Proposition 2: the optimization still favors memorization when possible. Larger datasets simply make perfect memorization less feasible, not less likely as an objective.

---

Simulated 1-ReFlow: CFM Memorizes Its Own Outputs

Setup: To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.

• Initial training: We train a base CFM model ($\sigma = 0$) on random, non-OT pairings.
• Integration: Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.

Interpretation:
The results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\sigma = 0.05$), despite only a small amount of noise, avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.

---

References

[1] Korotin, Alexander, et al. Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark. NeurIPS 34 (2021): 14593–14605.

We thank the reviewer for the thoughtful feedback and for highlighting the need for clearer conceptual connections. In our revision, we will improve the paper’s structure and include new CelebA experiments (which are also detailed later in this rebuttal) that further validate our theoretical findings.

---

Addressing Weaknesses

"significance of the paper’s main results are not clear." , "The writing style [..] tends to assume that the reader should 'just get' why their problem is important"

Our paper sheds light on the underlying causes of common failure modes behind the state-of-the-art fixed-point generative models. Our main claim is that in noiseless transport settings, deterministic pairings and interpolants, low gradient variance does not imply $W_2$ optimality. This challenges three common assumptions:

1. Gradient variance arises from interpolant intersections [3, 4]
2. ReFlow [5] corrects such intersections via iteration
3. Convergence of ReFlow is well-understood [6]

While deterministic ReFlow can straighten paths, it lacks convergence guarantees. A now-retracted claim (with the authors highlighting flaws in the proof of this claim) [1] suggested two iterations suffice; we show that even after two steps, ReFlow may fail to yield injective couplings (Counterexample 1).

Building on this, we reframed Rectified Flows as an iterative process on finite datasets, as encountered in practice. This revealed that due to discretization and limited data, vector fields can skip over interpolant intersections (Figure 6), failing to correct suboptimal paths. Rather than converging to the optimal transport solution, repeated rectification can lead to memorization: the model simply reproduces the observed pairings, regardless of their quality. We formalize this in Proposition 2, showing that ReFlow can achieve zero loss and gradient variance by memorizing suboptimal deterministic couplings.

"Additional empirical validation"

We now include new experiments on CelebA OT pairs from [2] that directly validate Proposition 2 in both adversarial and ReFlow-style settings.

an overall lack of “connective tissue” in the writing.

We acknowledge the reviewer’s concern, and will strengthen the paper’s narrative thread by clarifying the following schema, both textually and through an improved Figure 1:

(A) Lemma 1 (Gaussian closed-form) → Prop. 1 (grad. variance bounds)
(B) Gaussian experiments
(A)+(B) ⇒ Conclusions (low grad. ≠ OT; reflow skips intersection) → (generalizes) → Prop. 2 (finite data).

Questions

Rotations in L137:

The role of rotation (L137) is to introduce a formulation for rotational vector field, which we then study the variance of in Proposition 1.

What Lemma 1 has to do with the central question that the paper is posing:

The main purpose of Lemma 1 is to give the closed-form OT vector field formula on which Proposition 1 builds. The extra fact about exact representation is not necessary in the proof of the Proposition and could be introduced as a following remark.

Significance of this proposition:

Proposition 1 supports our empirical findings, which show that low gradient variance does not imply optimality.

Section 4 relation to the analysis:

Section 4 extends our findings beyond Gaussians. Proposition 2 proves that for deterministic pairings, CFM can reach zero loss and gradient variance by memorizing, even when the vector field is not optimal.

This supports our central claim: low gradient variance is not a reliable proxy for transport quality. We clarify this thread throughout the paper.

---

New Experiments on CelebA

To empirically validate Proposition 2, we evaluate Conditional Flow Matching (CFM) on CelebA using ground-truth OT pairings from Korotin et al. [2]. Our goal is to test the central claim: when trained with deterministic interpolants, CFM is optimized to memorize its input pairings.

To test this, we compare:

• CFM (σ = 0): trained with deterministic interpolants
  $x_t = (1 - t) x_0 + t x_1$

• CFM (σ = 0.05): trained with slightly noisy interpolants
  $x_t = (1 - t) x_0 + t x_1 + \sqrt{t(1 - t)} \, \sigma \epsilon$

The small amount of noise ($\sigma = 0.05$) is not introduced for performance reasons but to break the injectivity between $(x_t, t)$ and $(x_0, x_1)$—disabling the memorization pathway outlined in Proposition 2. Both models use identical U-Net architectures.

---

Adversarial Pairings: CFM Optimizes for Memorization with Deterministic Interpolants

Setup: To directly test whether CFM with deterministic interpolants ($\sigma = 0$) is optimized to memorize suboptimal pairings, we construct an adversarial dataset:

• We start from ground-truth OT pairs $(x_0, x_1)$ derived from the benchmark of Korotin et al. [2].
• We shuffle the targets $(x_1)$, breaking the optimal structure and producing random pairings $(x_0, S(x_1))$ with no coherent transport semantics.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, S(x_1))$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the shuffled training targets to assess memorization.

Since the pairings are deliberately incorrect, a low L2 error to shuffled targets indicates overfitting to suboptimal data. High L2 to these targets signals resistance to memorization.

Interpretation:
The results directly support Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) minimizes loss by tightly memorizing its training targets, even when they are arbitrarily shuffled and non-optimal. In contrast, CFM with small noise ($\sigma = 0.05$) resists this memorization trap and generalizes significantly better to the true OT targets.

Extra comments:

• Memorization decreases with larger datasets, as expected, since model capacity is fixed, it becomes harder to memorize all pairings in the 50K setting.
• However, this does not contradict Proposition 2: the optimization still favors memorization when possible. Larger datasets simply make perfect memorization less feasible, not less likely as an objective.

---

Simulated 1-ReFlow: CFM Memorizes Its Own Outputs

Setup: To evaluate whether iterative flow models suffer from self-reinforcing memorization, we simulate a 1-step ReFlow scenario—a core case of Proposition 2.

• Initial training: We train a base CFM model ($\sigma = 0$) on random, non-OT pairings.
• Integration: Using this model, we generate new endpoints $(x_0, x^1)$ via ODE integration.

We compare CFM (σ = 0) and CFM (σ = 0.05) on $(x_0, x^1)$.

Evaluation Protocol (Lower is better):
We report L2 reconstruction error to the true OT targets to assess generalization, and the generated training targets $(x_0, x^1)$ to assess memorization. Since the pairings come from a previous CFM model trained on random (non-OT) data, a low L2 to $x^1$ indicates that the model is overfitting to flawed past outputs. High error rates to these targets signal that the model is resisting memorization and potentially correcting earlier mistakes.

Interpretation:
The results in the table provide further support for Proposition 2: CFM with deterministic interpolants ($\sigma = 0$) is highly effective at memorizing the training couplings, even when those pairings are suboptimal and self-generated. In contrast, CFM ($\sigma = 0.05$)—despite only a small amount of noise—avoids this memorization trap and learns to generalize toward the true OT map, despite never being directly trained on it.

---

References

[1] Saptarshi Roy, Vansh Bansal, Purnamrita Sarkar, and Alessandro Rinaldo. 2-rectifications are enough for straight flows: A theoretical insight into Wasserstein convergence.

[2] Korotin, Alexander, et al. Do neural optimal transport solvers work? A continuous Wasserstein-2 benchmark. NeurIPS 34 (2021): 14593–14605.

[3] Fjelde, T., Mathieu, E., and Dutordoir, V. An Introduction to Flow Matching. MLG Blog, January 2024.

[4] Gagneux, A., Martin, S., Emonet, R., Bertrand, Q., and Massias, M. A Visual Dive into Conditional Flow Matching. ICLR Blogposts, April 2025.

[5] Liu, Xingchao, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv:2209.03003 (2022).

[6] Lee, Sangyun, Zinan Lin, and Giulia Fanti. Improving the training of rectified flows. NeurIPS 37 (2024): 63082–63109.

We acknowledge the reviewer’s concern, and will strengthen the paper’s narrative thread by clarifying the following schema, both textually and through an improved Figure 1:

(A) Lemma 1 (Gaussian closed-form) → Prop. 1 (grad. variance bounds)
(B) Gaussian experiments
(A)+(B) ⇒ Conclusions (low grad. ≠ OT; reflow skips intersection) → (generalizes) → Prop. 2 (finite data).