Title: On the Relation between Rectified Flows and Optimal Transport
Venue: NeurIPS 2025
Accepted: yes
Keywords: rectified flows

optimal transport

flow matching
Abstract: This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counterexamples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper re-examines the relationship between rectified flows and optimal transport (OT), arguing decisively that rectified flows, even with gradient constraints, are not a reliable method for computing OT solutions. The authors substantiate this claim through counterexamples, including a key scenario where distributions on disconnected supports cause a non-optimal transport plan to become a fixed point of the iterative rectification process. They further show that a \"non-rectifiable\" coupling can achieve zero loss without being optimal, demonstrating that a vanishing training objective does not guarantee optimality. Beyond these refutations, the paper provides a foundational analysis of rectified flows—deriving invariance properties, giving explicit solutions for Gaussian cases, and establishing conditions for rectifiability. Finally, to address the identified issues with ill-posed dynamics, the authors propose a \"smoothed rectification\" procedure that injects noise at each iteration to ensure the transport problem remains well-defined.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The core contribution—the systematic refutation of the equivalence between rectified flows and OT—is original. The idea of linking failure modes to disconnected supports and non-rectifiable couplings is a novel perspective. \n\n2. The paper has good technical quality. The arguments are backed by mathematical proofs and precisely constructed counterexamples.\n\n3. The paper has good contribution. By demonstrating the limitations of using rectified flows for optimal transport, it may clarify misconceptions and prevents the community from pursuing a flawed direction.\n\nWeaknesses\n1. While the proposed \"smoothed rectification\" method is a valuable idea, its analysis feels preliminary. The theoretical guarantees are shown for the specific case where the source distribution is Gaussian, which may not cover all use cases. More importantly, while the authors prove that their method maintains a good convergence rate for the *loss*, they do not investigate if it actually helps in converging to the *true optimal transport plan*, especially in the challenging cases (like disconnected supports) that they identify.\n\n2. While the counterexample in Section 4.1 based on disconnected supports is technically sound and arguably the paper's most critical contribution, its practical significance is not immediately obvious from the text. The authors should explicitly state why this scenario is not just a theoretical curiosity but a common and fundamental challenge in practice.\n\n3.No experiments demonstrate the proposed smoothed rectification procedure.\n\nQUESTIONS:\nDoes noise injection also help the iterative process escape these non-optimal fixed points? For instance, in your disconnected support counterexample (Fig. 1), if one were to run the smoothed rectification procedure starting from the independent coupling, would it be more likely to converge to the globally optimal transport plan (Fig. 1a) compared to the standard procedure? My evaluation of the paper's practical contribution would be further strengthened if the authors could provide either (a) an empirical study, perhaps by extending the experiments in Appendix B, to show that smoothing helps find the correct OT plan, or (b) a theoretical argument or conjecture on how noise might provide a regularizing effect that favors the true optimum.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors of this paper focus on further study of rectified flows, and in particular their relation to optimal transport. Rectified flows straighten the paths of a coupling (transport plan in the language of optimal transport) between two distributions. Given that the paths generated by the velocity field of an optimal transport (in the euclidean case) are straight, the goal is to learn an optimal transport through this rectification. The authors show by providing counterexamples that the assumptions for this to hold are stronger than in a previous paper. They also prove invariances for rectified flows under affine transformations, find formulas for rectified flows (which are optimal transports) in the gaussian case, and propose to smooth the starting distribution at each rectification step to ensure that the couplings stay rectifiable.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Many theoretical contributions to the study of rectified flows\n\nWeaknesses:\n1. To improve clarity, Definitions should be included. For example, the rectified coupling is defined on line 110 with a reference to equation (1). I think the paper would have been easier to follow if such important definitions were in Definition sections such as theorems and propositions.\n\nQUESTIONS:\n1. Isn't there a mistake in the definition (12) of $X_1$ in Remark 9? Shouldn't the second case also contain $X_0$ instead of $X_1$?\n2. To make sure that I understand the smoothed rectification in Theorem 15, you assume $\\mu_0$ is gaussian, and since rectification preserves the marginals, $X_0$ will remain gaussian, but it is the conditional distribution of $X_0$ (given $X_1$) that must be absolutely continuous with a smooth and positive density, which is why you add an independent Gaussian noise?\n\nLIMITATIONS:\n1. Counterexamples are provided which mean that stronger assumptions are needed for rectification to converge to an optimal transport, but these stronger assumptions are not investigated. This is of course minor as I think the paper is rich enough in results.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors clarify the relationship between couplings obtained from rectified flows and optimal transport in the setting of vector fields induced by potentials. In particular, they demonstrate via counter-examples, that fixed points and zero-minimisers of the rectified flow objective do not necessarily obtain optimal couplings. This corrects a previous result in the literature which claims the opposite. Moreover, the authors clarify conditions when solutions to the rectified flow objective are indeed rectifiable, and propose a procedure for ensuring rectifiability.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- While the main contribution presented in the paper is probably already known to theorists in this area, due to the simplicity of the counter-examples presented in the paper, the paper is still a nice addition to the literature for providing a single citable reference, for newcomers to the area, in clarifying the extent to which rectified flows can recover optimal transport (as well as correcting a previous mistake in the literature). \n\nWeaknesses:\n\n- The structure of the paper is a bit confusing. For instance, what is the purpose of Section 3.1? This section appears abruptly from the previous discussion of rectified flows with the gradient potential conditions, and appears to have little connection with the rest of the presentation. From inspection, it appears that Section 3.1 is only used in the proofs of later results. The layout disrupts from the presentation of the main argument of the paper and makes the paper harder to follow.  \n- Moreover, the paper is a bit overstuffed with off-hand remarks of results which have no accompanying justification. For example:\n    - Line 193: \"$v^{A,b}_t$ has a potential if and only if the Jacobian is symmetric.\" - what is the justification of this? I'm not sure how this immediately follows?\n    - Line 244: \"The statement of...\". The full statement of the corrected proposition and proof would be valuable to include in this paper to resolve any ambiguity, instead of leaving this offhand remark halfway in the paper  \n- For Theorem 2 and Theorem 4, covariance matrices are referred to as both $\\Sigma_0$ and $\\Sigma_{00}$. Why is this inconsistent? \n- In line 184, I do not understand how the independent coupling of $(X_0, X_1)$ follows a GMM distribution as $X_1$ follows a discrete measure. Hence, I do not see how Theorem 4 can be applied to this case.\n- The notation in Remark 5 is confusing. I assume that here $\\varphi_t^a$ is referring to the transformation by adding the constant $a$. However, in Theorem 1, the notation for this transformation uses the letter $b$ in $v_t^b$. I don't see any reason for changing up the notation here, apart from to reduce clarity.\n\nQUESTIONS:\nTo what extend can the counter-examples/proofs/intuition from the paper be generalised to rectified flows under the setting of a generic cost function, as presented in [1]?\n\n[1] Qiang Liu (2022) Rectified Flow: A Marginal Preserving Approach to Optimal Transport\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}