Pith. sign in

REVIEW 4 major objections 4 minor 64 references

Computational and Statistical Guarantees of the \textit{c}-Rectified flow

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read The paper proves that iterating c-rectified flow—a cost-aware variant that projects velocities onto gradient fields while preserving marginals—converges to the optimal transport coupling under compactness and uniform-integrability assumptio

desk verdict Real theoretical contributions in the qualitative convergence theorem and score minimax rates, but the headline 'always converges' is conditional on regularity assumptions the paper never derives, and the statistical claim trades on an exact plug-in rather than the iterative algorithm. read the letter →

arxiv 2608.02487 v1 pith:SCHY7AQK submitted 2026-08-03 stat.ML cs.LGmath.OCmath.PRmath.STstat.TH

Leda Wang , Zhehao Xu , Qiang Liu , Harrison H. Zhou This is my paper
classification stat.MLcs.LGmath.OCmath.PRmath.STstat.TH MSC 49Q2262G0568T07
keywords optimaltransportrectifiedflowc-rectifiedstatisticalestimationconvergenceratesscoreminimaxWassersteindistance
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

This paper tries to establish that a cost-aware variant of rectified flow, called c-rectified flow, is a principled algorithm for optimal transport. Ordinary rectified flow learns a velocity field by regression and can be iterated, but the paper shows in a Gaussian example that it can stabilize at a deterministic coupling that is not the optimal transport map unless the source and target covariance matrices commute. The proposed variant projects the learned velocity onto a gradient class compatible with the transport cost while preserving endpoint marginals; under compactness and uniform-integrability assumptions, the paper proves that iterating this projection drives the transport cost to the optimal value and the couplings converge weakly to the unique optimal coupling. Under additional projection-stability conditions, the convergence is quantitative: one step removes a fixed fraction of the excess cost, and iteration gives exponential decay, for both quadratic and strongly convex displacement costs. On the statistical side, the paper develops minimax-optimal score estimators for a Gaussian-ratio model and shows that plugging them into iterated c-rectified flow yields a rate-optimal estimator of the optimal transport map for d≥3 and a nearly parametric rate for d=1,2.

What carries the argument

Central object: the c-rectified flow update. From a coupling (X0,X1), take the linear interpolation X_t=(1−t)X0+tX1 and minimize L_{X,c}(f)=∫₀¹ E[m_c(Ẋ_t,∇f_t(X_t))]dt, where m_c is the Bregman divergence of the convex cost c; the new process follows dZ_t=∇c*(∇f_t(Z_t))dt. In the quadratic case, this is orthogonal projection of the conditional velocity onto gradient fields, i.e. removal of the divergence-free part. The key identity: a fixed point of this iteration has L=0 exactly for c-optimal couplings, so any subsequential limit with L=0 is the unique optimal coupling. Contraction rests on projection stability—a fixed fraction of T−T* must lie in the divergence-free subspace—which in the G

What would settle it

Run the population c-rectified iteration on centered Gaussian marginals N(0, Σ1), N(0, Σ2) with Σ1 and Σ2 chosen so that Σ1Σ2 ≠ Σ2Σ1 (say, Σ1=diag(4,1), Σ2 a rotation of it). The paper predicts the excess cost E||Z1^k−Z0^k||²−W₂²(P,Q) decays exponentially. If it stays bounded away from zero after many iterations, Theorem 1's convergence claim fails.

Watch

Extended reading notes

Core claim

c-rectified flow is claimed to be the right algorithmic object: unlike ordinary rectified flow, which in Gaussian cases can stabilize at a suboptimal coupling unless covariance matrices commute, the cost-aware gradient projection makes iteration converge to the optimal transport coupling. Under uniform Hölder and uniform-integrability assumptions, the transport cost along iterates converges to W₂²(P,Q) and the couplings converge weakly to the unique Brenier coupling. Under a projection-stability condition, one step contracts the excess cost by a fixed factor, giving exponential decay, for quadratic and general strongly convex displacement costs. The paper also supplies minimax score-estimati

Load-bearing premise

Everything rests on the assumption that, at every iteration, the smoothness of the rectified potentials and the growth of their gradients stay uniformly controlled, with the gradient integrability along the interpolated paths uniform over all steps; if those bounds fail, the proof cannot extract a limiting potential or show the limiting coupling is optimal.

Editorial extensions

If this is right

  • Iterating c-rectified flow is a principled population-level method for optimal transport: the transport cost converges to the optimal value and the coupling converges weakly to the unique optimal coupling.
  • Vanilla rectified flow, without cost-aware gradient projection, can stabilize at a non-optimal deterministic Gaussian coupling; practitioners should not expect reflow alone to recover optimal transport in general.
  • Under projection stability, the excess transport cost decays exponentially, so the number of iterations needed for a given accuracy can be bounded explicitly.
  • Combined with the paper's score-based marginal estimators, c-rectified flow yields a plug-in optimal transport map estimator with squared L2 risk at the minimax rate for d≥3 and nearly parametric in d=1,2.
  • The general-cost analogue shows the same convergence and contraction for strongly convex displacement costs, so the results are not tied to squared Euclidean distance.

Reading between the lines

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

  • Editorial inference: the qualitative convergence theorem likely extends to any setting with a unique optimal coupling and uniform compactness of the potentials; non-Gaussian marginals with log-concave densities are natural candidates, though the paper does not prove this.
  • Editorial inference: the projection-stability condition suggests a practical stopping rule—monitor the size of the divergence-free component of the velocity; when it vanishes, the iterates have reached the optimal fixed point.
  • Editorial inference: the Gaussian counterexample implies that any implementation of vanilla reflow on noncommuting covariance data silently converges to a suboptimal map; a cost-aware projection is therefore not an optional refinement but a necessary correction.
  • Editorial inference: since the plug-in analysis only uses Wasserstein control of the marginals, replacing the score-based marginal estimator with any other rate-optimal marginal estimator would transfer the same optimal transport map rate to c-rectified flow.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the c-rectified flow, a cost-aware variant of rectified flow that projects the learned velocity onto a gradient class while preserving endpoint marginals. Its main claims are: (i) ordinary rectified flow can stabilize at a non-optimal Gaussian coupling when the source and target covariance matrices do not commute; (ii) under compactness and uniform-integrability hypotheses, iterative c-rectified flow converges weakly to the optimal transport coupling and the transport cost converges to the optimal value; (iii) under projection-stability assumptions the iteration contracts the excess cost geometrically, both for quadratic and for strongly convex displacement costs; and (iv) a new score-estimation minimax theory yields plug-in estimators of optimal transport maps at rates that are claimed to be optimal for d>=3 and nearly parametric for d=1,2. The paper contains detailed proofs of the Gaussian calculations, explicit matrix formulas for the contractions, and matching upper and lower bounds for score estimation in the Gaussian-ratio model.

Significance. If the main structural claims are fully established, the paper would be a valuable contribution: it identifies a clear failure mode of vanilla rectified flow, proposes a principled cost-aware correction, and provides quantitative and statistical guarantees that are rare in this area. The Gaussian counterexample (Proposition 2) is clean and checkable, the Gaussian projection calculations in Section E are useful, and the score-estimation lower bounds in Appendix H are substantial. However, the central qualitative convergence theorem is conditional on regularity and uniform-integrability hypotheses whose propagation under the c-rectified update is not proved, and the quantitative contraction theorems require uniform projection stability that is only verified locally. The statistical optimality claim likewise relies on oracle operations rather than on a finite-step analysis of the iterative algorithm. The significance of the paper is therefore real but currently more prospective than established.

major comments (4)
  1. [Theorem 1 and Appendix D] The main qualitative claim is made conditional on two hypotheses: a uniform C^{2+alpha,1+alpha} bound for the minimizers f^{Z(k)} on every ball, and uniform integrability of the family of integrals of g(Z_t^k)^2. These hypotheses are assumed for every k, but no lemma or proposition shows that they are propagated by the c-rectified update from the initial coupling. The proof uses them to run the diagonal Arzela-Ascoli argument and to pass E[V_k] to E[V_*] via Lemma 5; without them the subsequential limit f* and the identity L_{Z*}(f*)=0 are not justified. The only verification is the Gaussian/linear case in Section E, where the potentials are quadratic. The abstract's formulation that iterative c-rectified flow 'always converges' is therefore too strong as stated. This is the load-bearing step for the qualitative convergence claim and needs either a propagation theorem for natural classes
  2. [Theorems 4-5 and 10-11; Assumptions 3 and 5] The exponential convergence results require Assumption 3 (resp. Assumption 5) to hold for every iterate T_k with the same constants. This is not derived from the one-step contraction theorem; it is an additional uniform structural assumption. Proposition 4 proves Assumption 3 only along smooth P-measure-preserving perturbations of the identity near T*, not that the iterates remain in the neighborhood where such a bound holds. Remark 1 acknowledges that entry into the admissible class is 'expected to occur' but no proof is given. Consequently, the advertised quantitative convergence guarantees are not established for arbitrary initial couplings; they are conditional on a condition that is verified only locally.
  3. [Section 4.2 and Algorithm 3] The statistical estimator in Algorithm 3 requires, in Step 4, the exact optimal transport map between the projected estimated marginals, and the parenthetical 'equivalently the limiting map obtained by Algorithm 1' invokes the iterative c-rectified flow. However, Theorems 1 and 9 provide only asymptotic convergence and only under assumptions that are not verified for the plug-in marginals; no finite-K error bound is given. Thus the stated risk bound O(rho_n^2) in Proposition 5 is an oracle plug-in bound, not a guarantee for an estimator obtained by running finitely many steps of Algorithm 1. To support the claim that iterative c-rectified flow yields a rate-optimal estimator, the paper needs either a finite-iteration bias analysis or an explicit statement that the statistical theorem concerns the exact OT map and only the asymptotic guarantee of the iterative approximation.
  4. [Section 4.2, Proposition 5] The text describes the resulting estimator as 'rate-optimal' for optimal transport map estimation, but no matching lower bound for the map risk is proved in the manuscript. Theorem 6 gives minimax lower bounds only for the score-estimation problem; Theorem 7 gives upper bounds for the Wasserstein marginal risk; Proposition 5 transfers that upper bound to the map risk. To justify the word 'rate-optimal', the authors need either to prove a lower bound for the map estimation problem under the same class, or to cite precisely and verify the hypotheses of an existing minimax lower bound for OT maps (for instance, the smooth-Brenier-map lower bound of Hutter-Rigollet or Manole et al.) and show that the implied map smoothness is inherited from F_{alpha,d} and C^lambda_Lambda. As it stands, the optimality claim is an assertion rather than a proved consequence.
minor comments (4)
  1. [Notation, Section 1.4] The symbol P is used both for a probability measure (e.g. P(A), P(X)) and for the set of probability measures P(A). This overload is confusing in several places and should be disambiguated.
  2. [Theorem 1] The phrase 'Denote (Z*_0,Z*_1) as the unique optimal coupling' introduces uniqueness as an assumption, but it is not listed among the theorem hypotheses. Since uniqueness is needed to pass from subsequential convergence to full weak convergence, it should be stated explicitly as an assumption.
  3. [Theorem 5] Typo: 'a initial coupling' should be 'an initial coupling'. There are similar minor grammar issues elsewhere in Section 3.3.
  4. [Equation (10)] The notation '1 ∨ log_+(nt^{alpha+d/2})^{d/2}' is ambiguous: the exponent d/2 appears to apply to the whole logarithm expression in the first displayed branch, but the later derivations treat it as (log(...))^{d/2}. Clarify the convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: convergence and rates are derived from explicit assumptions; the imported fixed-point theorem is independent support, not a restatement of the target result.

full rationale

The derivation chain is not circular. Theorem 1 obtains convergence by a tightness/subsequence argument: cost monotonicity gives a decreasing sequence, the c-rectified objective tends to zero along a subsequence, and the Arzelà–Ascoli diagonal argument uses the assumed uniform C^{2+α,1+α} bounds to pass to a limit potential; the uniform-integrability hypothesis is used via Lemma 5 to pass E[V_k] → E[V_*]. The final identification of the limit as the optimal transport coupling invokes Theorem 16 of Liu (2022), a fixed-point characterization asserting that L=0 iff the coupling is c-optimal. This is an imported theorem from prior work by a coauthor and is load-bearing, but it is a parameter-free statement about stationary points, not a restatement of the convergence claim, and it does not depend on the present paper's fitted values. Thus it counts as independent support rather than circularity. The quantitative contraction results (Theorems 2–5, 10–11) are proved through explicit projection bounds (Gaussian Helmholtz decomposition, spectral criteria, transported divergence-free fields) and separate cost-difference lemmas; no parameter is tuned and then renamed as a prediction. The statistical rates in Theorems 6–7 are derived from matching upper and lower bounds, and the plug-in optimal transport risk follows from external stability estimates (Theorem 8 of Balakrishnan and Manole 2025 and the paper's Theorem 12). The main caveat is that Theorem 1's uniform Hölder and uniform-integrability assumptions are not shown to propagate along the iterates for general P,Q; this is a conditionality/rigor gap, not a circular reduction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or model entities. Its free parameters are standard tuning choices. The main burden is carried by regularity and projection-stability conditions on the marginals and the transport maps, plus the external stationary-point characterization from Liu (2022).

free parameters (1)
  • local-polynomial bandwidth h_n = n^{-1/(2α+d)}
    Chosen by bias-variance balancing in Proposition 8 and the score estimator; a tuning choice, not fitted to the target distribution or to the OT map.
assumptions (6)
  • standard math Standard optimal transport facts: Brenier's theorem, Kantorovich duality, Benamou-Brenier formula, Caffarelli contraction.
    Invoked throughout Sections 2 and 4.2 (Theorem 13, equation (14), and Theorem 8/cited stability result).
  • standard math Stationary characterization of Liu (2022): for c-rectifiable X, (X0,X1)=c-Rectify((X0,X1)) iff L_{X,c}(f)=0 iff the coupling is c-optimal.
    Used as Theorem 16 in Appendix B.5.0.3 to identify the weak limit in Theorem 1 as the OT coupling; taken from a coauthor's prior paper and not reproved here.
  • domain assumption Uniform C^{2+α,1+α} bounds and uniform integrability of the gradient process in Theorem 1.
    Assumptions in Theorem 1 needed for Arzelà-Ascoli and Vitali arguments; not verified for the statistical models.
  • domain assumption Assumption 3 (projection stability): a fixed fraction of T-T* must lie in the p-divergence-free subspace.
    Drives Theorems 4-5; only proven locally near T* in Proposition 4, and required uniformly along all iterates in Theorem 5.
  • domain assumption Densities belong to the Gaussian-ratio Hölder class F_{α,d}; marginals in C^λ_Λ for stability.
    Needed for minimax score rates (Theorem 6) and plug-in map stability (Proposition 5).
  • domain assumption Unique optimal coupling and c-rectifiability of the iterates.
    Stated in Theorems 1, 9, and 5; without uniqueness, the weak-convergence conclusion is not justified.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Computational and Statistical Guarantees of the \textit{c}-Rectified flow." pith.science (2026). https://pith.science/paper/SCHY7AQK

@misc{pith2026260802487,
  author       = {Pith},
  title        = {Pith review of: Computational and Statistical Guarantees of the \textitc-Rectified flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCHY7AQK}},
  note         = {Machine review of arXiv:2608.02487}
}
abstract

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying \textit{c}-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative \textit{c}-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a H\"older ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative \textit{c}-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

64 extracted references · 12 linked inside Pith

  1. [1]

    Communications in Mathematical Physics , volume=

    Monotonicity properties of optimal transportation and the fkg and related inequalities , author=. Communications in Mathematical Physics , volume=. 2000 , publisher=

    2000

  2. [2]

    arXiv preprint arXiv:2604.22670 , year=

    Curvature of optimal transport with respect to the cost and applications to inverse optimal transport , author=. arXiv preprint arXiv:2604.22670 , year=

    arXiv

  3. [3]

    arXiv preprint arXiv:2502.12326 , year=

    Stability bounds for smooth optimal transport maps and their statistical implications , author=. arXiv preprint arXiv:2502.12326 , year=

    arXiv

  4. [4]

    Proceedings of the 39th International Conference on Machine Learning , series =

    Debiaser Beware: Pitfalls of Centering Regularized Transport Maps , author =. Proceedings of the 39th International Conference on Machine Learning , series =. 2022 , publisher =

    2022

  5. [5]

    Proceedings of the 40th International Conference on Machine Learning , series =

    Minimax Estimation of Discontinuous Optimal Transport Maps: The Semi-Discrete Case , author =. Proceedings of the 40th International Conference on Machine Learning , series =. 2023 , publisher =

    2023

  6. [6]

    The Annals of Statistics , volume =

    Optimal Transport Map Estimation in General Function Spaces , author =. The Annals of Statistics , volume =. 2025 , doi =

    2025

  7. [7]

    International Mathematics Research Notices , volume =

    Tight Stability Bounds for Entropic Brenier Maps , author =. International Mathematics Research Notices , volume =. 2025 , doi =

    2025

  8. [8]

    Probability Theory and Related Fields , volume =

    On the Rate of Convergence in Wasserstein Distance of the Empirical Measure , author =. Probability Theory and Related Fields , volume =

Show all 64 references
  1. [9]

    Bernoulli , volume =

    Sharp Asymptotic and Finite-Sample Rates of Convergence of Empirical Measures in Wasserstein Distance , author =. Bernoulli , volume =

  2. [10]

    Journal of the Royal Statistical Society: Series B , volume =

    Inference for Empirical Wasserstein Distances on Finite Spaces , author =. Journal of the Royal Statistical Society: Series B , volume =

  3. [11]

    The Annals of Probability , volume =

    Central Limit Theorems for Empirical Transportation Cost in General Dimension , author =. The Annals of Probability , volume =

  4. [12]

    The Annals of Statistics , volume =

    Minimax Estimation of Smooth Densities in Wasserstein Distance , author =. The Annals of Statistics , volume =

  5. [13]

    Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics , pages =

    Sample Complexity of Sinkhorn Divergences , author =. Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics , pages =

  6. [14]

    Advances in Neural Information Processing Systems , year =

    Statistical Bounds for Entropic Optimal Transport: Sample Complexity and the Central Limit Theorem , author =. Advances in Neural Information Processing Systems , year =

  7. [15]

    arXiv preprint arXiv:2109.12004 , year =

    Entropic Estimation of Optimal Transport Maps , author =. arXiv preprint arXiv:2109.12004 , year =

    arXiv

  8. [16]

    Electronic Journal of Statistics , volume =

    Limit Theorems for Entropic Optimal Transport Maps and Sinkhorn Divergence , author =. Electronic Journal of Statistics , volume =

  9. [17]

    The Annals of Statistics , volume =

    On the Sample Complexity of Entropic Optimal Transport , author =. The Annals of Statistics , volume =

  10. [18]

    Nietert, Sloan and Goldfeld, Ziv and Kato, Kengo , booktitle =. Smooth

  11. [19]

    arXiv preprint arXiv:2502.01310 , year =

    A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers , author =. arXiv preprint arXiv:2502.01310 , year =

  12. [20]

    Numerische Mathematik , volume=

    A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , author=. Numerische Mathematik , volume=. 2000 , publisher=

    2000

  13. [21]

    A Survey of the Schr

    L. A Survey of the Schr. Discrete and Continuous Dynamical Systems - Series A , volume =

  14. [22]

    An Optimal Transport Approach for the Schr

    Di Marino, Simone and Gerolin, Augusto , journal =. An Optimal Transport Approach for the Schr

  15. [23]

    Probability Theory and Related Fields , volume =

    Entropic Optimal Transport: Convergence of Potentials , author =. Probability Theory and Related Fields , volume =

  16. [24]

    Duke Mathematical Journal , volume =

    Entropic Optimal Transport: Geometry and Large Deviations , author =. Duke Mathematical Journal , volume =

  17. [25]

    Diffusion Schr

    Shi, Yuyang and De Bortoli, Valentin and Campbell, Andrew and Doucet, Arnaud , journal =. Diffusion Schr

  18. [26]

    Diffusion Bridge Mixture Transports, Schr

    Peluchetti, Stefano , journal =. Diffusion Bridge Mixture Transports, Schr

  19. [27]

    Diffusion & Adversarial Schr

    Kholkin, Sergei and Ksenofontov, Grigoriy and Li, David and Kornilov, Nikita and Gushchin, Nikita and Suvorikova, Alexandra and Kroshnin, Alexey and Burnaev, Evgeny and Korotin, Alexander , journal =. Diffusion & Adversarial Schr

  20. [28]

    arXiv preprint arXiv:2510.20871 , year =

    Exponential Convergence Guarantees for Iterative Markovian Fitting , author =. arXiv preprint arXiv:2510.20871 , year =

  21. [29]

    arXiv preprint arXiv:2209.14577 , year=

    Rectified flow: A marginal preserving approach to optimal transport , author=. arXiv preprint arXiv:2209.14577 , year=

    arXiv

  22. [30]

    International Conference on Learning Representations , year =

    Flow Matching for Generative Modeling , author =. International Conference on Learning Representations , year =

  23. [31]

    arXiv preprint arXiv:2303.08797 , year =

    Stochastic Interpolants: A Unifying Framework for Flows and Diffusions , author =. arXiv preprint arXiv:2303.08797 , year =

    arXiv

  24. [32]

    International Conference on Learning Representations , year =

    Flow Matching Achieves Almost Minimax Optimal Convergence , author =. International Conference on Learning Representations , year =

  25. [33]

    arXiv preprint arXiv:2504.13336 , year =

    On the Minimax Optimality of Flow Matching through the Connection to Kernel Density Estimation , author =. arXiv preprint arXiv:2504.13336 , year =

  26. [34]

    arXiv preprint arXiv:2601.20250 , year =

    Order-Optimal Sample Complexity of Rectified Flows , author =. arXiv preprint arXiv:2601.20250 , year =

  27. [35]

    Transactions on Machine Learning Research , year =

    Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport , author =. Transactions on Machine Learning Research , year =

  28. [36]

    arXiv preprint arXiv:2304.14772 , year =

    Multisample Flow Matching: Straightening Flows with Minibatch Couplings , author =. arXiv preprint arXiv:2304.14772 , year =

    arXiv

  29. [37]

    arXiv preprint arXiv:2410.14949 , year =

    On the Wasserstein Convergence and Straightness of Rectified Flow , author =. arXiv preprint arXiv:2410.14949 , year =

    arXiv

  30. [38]

    Proceedings of the 37th International Conference on Machine Learning , pages =

    Optimal Transport Mapping via Input Convex Neural Networks , author =. Proceedings of the 37th International Conference on Machine Learning , pages =

  31. [39]

    The Annals of Statistics , volume =

    Plugin Estimation of Smooth Optimal Transport Maps , author =. The Annals of Statistics , volume =

  32. [40]

    2021 , publisher=

    Topics in optimal transportation , author=. 2021 , publisher=

    2021

  33. [41]

    Foundations and Trends in Machine Learning , volume=

    Computational Optimal Transport , author=. Foundations and Trends in Machine Learning , volume=

  34. [42]

    2009 , publisher=

    Optimal transport: old and new , author=. 2009 , publisher=

    2009

  35. [43]

    Cell , volume=

    Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming , author=. Cell , volume=. 2019 , publisher=

    2019

  36. [44]

    Journal of Computational Physics , volume=

    Bayesian inference with optimal maps , author=. Journal of Computational Physics , volume=. 2012 , publisher=

    2012

  37. [45]

    The Annals of Statistics , number =

    Minimax estimation of smooth optimal transport maps , author=. The Annals of Statistics , number =

  38. [46]

    arXiv preprint arXiv:2206.13472 , year=

    On the sample complexity of entropic optimal transport , author=. arXiv preprint arXiv:2206.13472 , year=

    arXiv

  39. [47]

    arXiv preprint arXiv:2209.03003 , year=

    Flow straight and fast: Learning to generate and transfer data with rectified flow , author=. arXiv preprint arXiv:2209.03003 , year=

    arXiv

  40. [48]

    International Conference on Learning Representations , year=

    InstaFlow: One Step Is Enough for High-Quality Diffusion-Based Text-to-Image Generation , author=. International Conference on Learning Representations , year=

  41. [49]

    Proceedings of the 41st International Conference on Machine Learning , series=

    Scaling Rectified Flow Transformers for High-Resolution Image Synthesis , author=. Proceedings of the 41st International Conference on Machine Learning , series=. 2024 , publisher=

    2024

  42. [50]

    arXiv preprint arXiv:2506.15742 , year=

    arXiv

  43. [51]

    Journal of Optimization Theory and Applications , volume=

    On the optimal mapping of distributions , author=. Journal of Optimization Theory and Applications , volume=. 1984 , publisher=

    1984

  44. [52]

    Bernoulli , volume=

    Estimation of wasserstein distances in the spiked transport model , author=. Bernoulli , volume=. 2022 , publisher=

    2022

  45. [53]

    arXiv preprint arXiv:2409.07032 , year=

    From optimal score matching to optimal sampling , author=. arXiv preprint arXiv:2409.07032 , year=

    arXiv

  46. [54]

    Journal of Machine Learning Research , volume=

    Estimation of non-normalized statistical models by score matching , author=. Journal of Machine Learning Research , volume=

  47. [55]

    Neural Computation , volume=

    A connection between score matching and denoising autoencoders , author=. Neural Computation , volume=. 2011 , publisher=

    2011

  48. [56]

    International Conference on Learning Representations , year=

    Score-Based Generative Modeling through Stochastic Differential Equations , author=. International Conference on Learning Representations , year=

  49. [57]

    arXiv preprint arXiv:2511.03193 , year=

    Statistical Properties of Rectified Flow , author=. arXiv preprint arXiv:2511.03193 , year=

  50. [58]

    arXiv preprint arXiv:2505.19712 , year=

    On the relation between rectified flows and optimal transport , author=. arXiv preprint arXiv:2505.19712 , year=

  51. [59]

    Journal of Machine Learning Research , volume=

    Derivative estimation with local polynomial fitting , author=. Journal of Machine Learning Research , volume=. 2013 , publisher=

    2013

  52. [60]

    Geometric & Functional Analysis GAFA , volume=

    Transportation cost for Gaussian and other product measures , author=. Geometric & Functional Analysis GAFA , volume=. 1996 , publisher=

    1996

  53. [61]

    A survey , author=

    Transport inequalities. A survey , author=. arXiv preprint arXiv:1003.3852 , year=

    arXiv

  54. [62]

    Stochastic Analysis, Filtering, and Stochastic Optimization: A Commemorative Volume to Honor Mark HA Davis's Contributions , pages=

    Optimal couplings on Wiener space and an extension of Talagrand’s transport inequality , author=. Stochastic Analysis, Filtering, and Stochastic Optimization: A Commemorative Volume to Honor Mark HA Davis's Contributions , pages=. 2022 , publisher=

    2022

  55. [63]

    The Annals of Probability , pages=

    Time reversal of diffusions , author=. The Annals of Probability , pages=. 1986 , publisher=

    1986

  56. [64]

    Comptes rendus des s

    Deux remarques sur l'estimation , author=. Comptes rendus des s

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.