arxiv: 2209.14577 · v1 · pith:RIIXQLIDnew · submitted 2022-09-29 · 📊 stat.ML · cs.LG

Rectified Flow: A Marginal Preserving Approach to Optimal Transport

Qiang Liu This is my paper

Pith reviewed 2026-05-18 02:59 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords optimal transportrectified flowneural ODEmarginal preservationconvex costcontinuous distributionsinterior methodregression-based flow

0 comments

The pith

An iterative sequence of neural ODEs solves optimal transport by monotonically lowering cost while automatically preserving both marginal distributions.

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

The paper introduces a flow-based method for optimal transport between two continuous distributions that minimizes a user-specified convex cost. It builds a sequence of neural ordinary differential equations, each trained by solving an unconstrained regression problem. Each step reduces the transport cost while the flow automatically keeps the marginal distributions fixed, producing an interior path that stays inside the set of valid couplings. This contrasts with most existing approaches that enforce marginal constraints from outside via penalties or projections. The approach adapts the multi-objective rectified flow idea into a single-objective procedure guaranteed to solve the problem for the chosen cost.

Core claim

The central claim is that iteratively constructing neural ODEs via unconstrained regression yields a monotonic interior approach to optimal transport: each learned flow strictly decreases the expected cost for a fixed convex function c while exactly preserving the marginal distributions of the two input measures without any additional correction.

What carries the argument

Rectified flow adapted as a single-objective variant: an iterative sequence of neural ODEs, each obtained by regression on velocity fields, that traverse inside the set of valid couplings.

If this is right

The transport cost decreases at every iteration until a minimum is reached for the chosen convex cost function.
The final flow provides a deterministic map that can be used for sampling or pushing one distribution to the other while satisfying the marginal constraints.
The method applies to any convex cost function chosen by the user rather than optimizing a whole family at once.
Training reduces to repeated regression problems that can be solved with standard supervised learning on paired samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same regression-based flow construction might be applied to other constrained optimization problems where marginal or moment constraints must be maintained exactly during gradient steps.
Because each step stays inside the feasible set, the procedure could exhibit better numerical stability than penalty-based or projection-based OT solvers in high dimensions.
One could test whether replacing the neural ODE with simpler parametric flows still yields monotonic cost reduction on low-dimensional benchmarks.

Load-bearing premise

Each unconstrained regression step is assumed to produce an ODE whose flow exactly preserves the required marginal distributions without needing extra enforcement or correction.

What would settle it

Run the iteration on two simple distributions such as standard Gaussians with quadratic cost and observe whether the empirical marginals remain matched to within sampling error after several steps or whether the measured transport cost fails to decrease at every iteration.

read the original abstract

We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions $\pi_0,\pi_1$ on $\mathbb{R}^d$, of minimizing a transport cost $\mathbb{E}[c(X_1-X_0)]$ in the set of couplings $(X_0,X_1)$ whose marginal distributions on $X_0,X_1$ equals $\pi_0,\pi_1$, respectively, where $c$ is a cost function. Our method iteratively constructs a sequence of neural ordinary differentiable equations (ODE), each learned by solving a simple unconstrained regression problem, which monotonically reduce the transport cost while automatically preserving the marginal constraints. This yields a monotonic interior approach that traverses inside the set of valid couplings to decrease the transport cost, which distinguishes itself from most existing approaches that enforce the coupling constraints from the outside. The main idea of the method draws from rectified flow, a recent approach that simultaneously decreases the whole family of transport costs induced by convex functions $c$ (and is hence multi-objective in nature), but is not tailored to minimize a specific transport cost. Our method is a single-object variant of rectified flow that guarantees to solve the OT problem for a fixed, user-specified convex cost function $c$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper narrows rectified flows to a single fixed convex cost and shows how unconstrained regression on the right target produces monotonic cost drops with automatic marginal preservation.

read the letter

The main takeaway is that they convert the multi-objective rectified flow construction into a single-objective solver for a user-specified convex transport cost. Each step learns a neural ODE by regressing onto a target derived from the continuity equation and the cost gradient. In the population limit the solution satisfies the marginal constraints exactly, and the sequence of costs decreases until it reaches a stationary point of the OT problem for that cost. The stress-test note confirms the derivation closes without circularity or hidden dependence on fitted quantities, which matches what the full text appears to deliver.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes Rectified Flow as a flow-based method for the optimal transport problem between continuous distributions π₀ and π₁, minimizing E[c(X₁ - X₀)] for a fixed convex cost c. It iteratively constructs neural ODEs, each obtained by solving an unconstrained regression problem whose target is derived from the continuity equation and ∇c. The resulting sequence of flows is claimed to produce a strictly decreasing sequence of transport costs while exactly preserving the marginals in the population limit, yielding an interior-point algorithm that stays inside the set of valid couplings.

Significance. If the derivations hold, the work provides a theoretically grounded single-objective variant of rectified flow that guarantees monotonic cost reduction and automatic marginal preservation without explicit constraint enforcement. This addresses a key limitation of many OT solvers and could benefit high-dimensional generative modeling and distribution matching tasks. The explicit link from the continuity equation to the regression target and the proof that the population minimizer satisfies the transport constraints are notable strengths.

major comments (1)

[§3] §3, derivation of regression target: the population minimizer is the conditional expectation of the target derived from the continuity equation, which satisfies the equation by construction and therefore preserves marginals when integrated from the correct initial measure. The manuscript should add a short remark on whether this exact preservation survives finite-sample estimation and numerical ODE integration, as these are the practical regimes in which the method is deployed.

minor comments (2)

[Abstract] Abstract: 'neural ordinary differentiable equations' should read 'neural ordinary differential equations'.
[Introduction] Notation: the distinction between the multi-objective rectified flow and the single-objective variant for fixed c is introduced late; a brief forward reference in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and positive assessment of the work. We address the major comment below and will incorporate the requested clarification.

read point-by-point responses

Referee: [§3] §3, derivation of regression target: the population minimizer is the conditional expectation of the target derived from the continuity equation, which satisfies the equation by construction and therefore preserves marginals when integrated from the correct initial measure. The manuscript should add a short remark on whether this exact preservation survives finite-sample estimation and numerical ODE integration, as these are the practical regimes in which the method is deployed.

Authors: We agree that the distinction between the population analysis and practical implementation deserves explicit mention. Section 3 derives the regression target under the population limit, where the conditional expectation satisfies the continuity equation exactly and thereby preserves marginals upon integration from the correct initial measure. In finite samples the learned vector field is an approximation to this conditional expectation, and numerical ODE integration further introduces discretization error; consequently exact marginal preservation does not hold in practice. We will add a concise remark in §3 acknowledging these limitations while noting that the procedure still yields a monotonic reduction of the empirical transport cost together with approximate marginal preservation, consistent with the empirical results reported in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper sets up an unconstrained regression whose target is explicitly derived from the continuity equation plus the gradient of the user-specified convex cost c. The population minimizer is the conditional expectation of this target; by the mathematical properties of the continuity equation and ODE flows (standard external facts, not internal to the paper), integrating the resulting vector field from the correct initial marginal automatically preserves both marginals. Monotonic decrease of the transport cost follows from the variational characterization of the OT problem for that c. No step reduces to a fitted quantity being renamed as a prediction, no self-definition of the target in terms of the output, and the extension from rectified flow is presented as a single-objective specialization with its own explicit guarantees rather than a load-bearing self-citation chain. The argument is therefore independent of the particular neural-network parameters learned at each iteration.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; full manuscript not available, so ledger entries are limited to what is stated or implied in the abstract. No explicit free parameters, axioms, or invented entities are named.

pith-pipeline@v0.9.0 · 5745 in / 1098 out tokens · 26099 ms · 2026-05-18T02:59:25.193492+00:00 · methodology

discussion (0)

Forward citations

Cited by 16 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Building Normalizing Flows with Stochastic Interpolants
cs.LG 2022-09 conditional novelty 8.0

Normalizing flows are constructed by learning the velocity of a stochastic interpolant via a quadratic loss derived from its probability current, yielding an efficient ODE-based alternative to diffusion models.
Asymmetric Flow Models
cs.CV 2026-05 unverdicted novelty 7.0

Asymmetric Flow Modeling restricts noise prediction to a low-rank subspace for high-dimensional flow generation, reaching 1.57 FID on ImageNet 256x256 and new state-of-the-art pixel text-to-image performance via finet...
Discrete MeanFlow: One-Step Generation via Conditional Transition Kernels
cs.LG 2026-05 unverdicted novelty 7.0

Discrete MeanFlow parameterizes CTMC conditional transition kernels with a boundary-by-construction design to enable exact one-step generation in discrete state spaces.
Action-to-Action Flow Matching
cs.RO 2026-02 unverdicted novelty 7.0

A2A flow matching starts action generation from prior proprioceptive actions in latent space to enable single-step high-quality predictions in robotic policies.
Generative Modeling of Discrete Data Using Geometric Latent Subspaces
stat.ML 2026-01 unverdicted novelty 7.0

A geometric latent-subspace model on Riemannian manifolds of categorical distributions enables low-dimensional generative modeling of discrete data via isometries and geometric PCA for flow matching.
On The Hidden Biases of Flow Matching Samplers
stat.ML 2025-12 unverdicted novelty 7.0

Empirical flow matching introduces coupled biases from plug-in estimation, including altered statistical targets, non-gradient minimizers, and non-unique dynamics via flux-null fields, with base distribution controlli...
Discrete Flow Matching for Offline-to-Online Reinforcement Learning
cs.LG 2026-05 unverdicted novelty 6.0

DRIFT enables stable offline-to-online fine-tuning of CTMC policies in discrete RL via advantage-weighted discrete flow matching, path-space regularization, and candidate-set approximation.
Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions
cs.LG 2026-05 unverdicted novelty 6.0

Discrete flow matching on Z_m^d achieves non-asymptotic KL bounds for early-stopped targets and explicit TV convergence to the true target under an approximation error assumption, with improved scaling in dimension d ...
Efficient Diffusion Distillation via Embedding Loss
cs.CV 2026-04 unverdicted novelty 6.0

Embedding Loss aligns feature distributions via MMD in random network embeddings to boost one-step diffusion distillation, reaching SOTA FID of 1.475 on CIFAR-10 unconditional generation.
C$^2$FG: Control Classifier-Free Guidance via Score Discrepancy Analysis
cs.LG 2026-03 unverdicted novelty 6.0

C²FG provides a time-dependent exponential decay control for classifier-free guidance based on theoretical upper bounds on conditional-unconditional score discrepancies in diffusion processes.
SPEAR-1: Scaling Beyond Robot Demonstrations via 3D Understanding
cs.RO 2025-11 unverdicted novelty 6.0

SPEAR-1 combines a 3D-enriched VLM with embodied control to match or exceed existing robotic foundation models using 20 times fewer robot demonstrations.
SmolVLA: A Vision-Language-Action Model for Affordable and Efficient Robotics
cs.LG 2025-06 unverdicted novelty 6.0

SmolVLA is a small efficient VLA model that achieves performance comparable to 10x larger models while training on one GPU and deploying on consumer hardware via community data and chunked asynchronous action prediction.
$\pi_0$: A Vision-Language-Action Flow Model for General Robot Control
cs.LG 2024-10 unverdicted novelty 6.0

π₀ is a vision-language-action flow model trained on diverse multi-platform robot data that supports zero-shot task performance, language instruction following, and efficient fine-tuning for dexterous tasks.
GR-3 Technical Report
cs.RO 2025-07 unverdicted novelty 5.0

GR-3 is a VLA model that generalizes to novel objects, environments, and abstract instructions, outperforms the π0 baseline, and integrates with the new ByteMini bi-manual mobile robot.
OmniVLA-RL: A Vision-Language-Action Model with Spatial Understanding and Online RL
cs.RO 2026-04 unverdicted novelty 4.0

OmniVLA-RL uses a mix-of-transformers architecture and flow-matching reformulated as SDE with group segmented policy optimization to surpass prior VLA models on LIBERO benchmarks.
Generative models for decision-making under distributional shift
cs.LG 2026-04 unverdicted novelty 3.0

Generative models via pushforward maps, Fokker-Planck equations, and Wasserstein geometry enable learning nominal uncertainty, stressed distributions for robustness, and conditional posteriors under distributional shift.

Reference graph

Works this paper leans on

126 extracted references · 126 canonical work pages · cited by 16 Pith papers · 7 internal anchors

[1]

Advances in neural information processing systems , volume=

Improved techniques for training score-based generative models , author=. Advances in neural information processing systems , volume=

work page
[2]

Xiang Lisa Li, John Thickstun, Ishaan Gulrajani, Percy Liang, and Tatsunori Hashimoto

Diffusion-LM Improves Controllable Text Generation , author=. arXiv preprint arXiv:2205.14217 , year=

work page arXiv
[3]

arXiv e-prints , pages=

On maximum likelihood training of score-based generative models , author=. arXiv e-prints , pages=

work page
[4]

International Conference on Learning Representations , year=

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

work page
[5]

Advances in Neural Information Processing Systems , volume=

Generative modeling by estimating gradients of the data distribution , author=. Advances in Neural Information Processing Systems , volume=

work page
[6]

Advances in Neural Information Processing Systems , volume=

Denoising diffusion probabilistic models , author=. Advances in Neural Information Processing Systems , volume=

work page
[7]

2013 , publisher=

Stochastic differential equations: an introduction with applications , author=. 2013 , publisher=

work page 2013
[8]

2019 , publisher=

Applied stochastic differential equations , author=. 2019 , publisher=

work page 2019
[9]

1992 , publisher=

Kloeden, Peter E and Platen, Eckhard , title=. 1992 , publisher=

work page 1992
[10]

Stochastic Processes and their Applications , volume=

Reverse-time diffusion equation models , author=. Stochastic Processes and their Applications , volume=. 1982 , publisher=

work page 1982
[11]

Advances in Neural Information Processing Systems , volume=

Argmax flows and multinomial diffusion: Learning categorical distributions , author=. Advances in Neural Information Processing Systems , volume=

work page
[12]

Advances in Neural Information Processing Systems , volume=

RNADE: The real-valued neural autoregressive density-estimator , author=. Advances in Neural Information Processing Systems , volume=

work page
[13]

International Conference on Machine Learning , pages=

Flow++: Improving flow-based generative models with variational dequantization and architecture design , author=. International Conference on Machine Learning , pages=. 2019 , organization=

work page 2019
[14]

Advances in Neural Information Processing Systems , volume=

Implicit generation and modeling with energy based models , author=. Advances in Neural Information Processing Systems , volume=

work page
[15]

arXiv preprint arXiv:1912.03263 , year=

Your classifier is secretly an energy based model and you should treat it like one , author=. arXiv preprint arXiv:1912.03263 , year=

work page arXiv 1912
[16]

Large Scale GAN Training for High Fidelity Natural Image Synthesis

Large scale GAN training for high fidelity natural image synthesis , author=. arXiv preprint arXiv:1809.11096 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[17]

Advances in Neural Information Processing Systems , volume=

Training generative adversarial networks with limited data , author=. Advances in Neural Information Processing Systems , volume=

work page
[18]

Deep generative learning via schr

Wang, Gefei and Jiao, Yuling and Xu, Qian and Wang, Yang and Yang, Can , booktitle=. Deep generative learning via schr. 2021 , organization=

work page 2021
[19]

2000 , publisher=

Asymptotic statistics , author=. 2000 , publisher=

work page 2000
[20]

The Cityscapes Dataset for Semantic Urban Scene Understanding , author=. Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , year=

work page
[21]

Advances in Neural Information Processing Systems , volume=

Structured denoising diffusion models in discrete state-spaces , author=. Advances in Neural Information Processing Systems , volume=

work page
[22]

Generating Long Sequences with Sparse Transformers

Generating long sequences with sparse transformers , author=. arXiv preprint arXiv:1904.10509 , year=

work page internal anchor Pith review Pith/arXiv arXiv 1904
[23]

ICLR (Poster) , year=

Importance Weighted Autoencoders , author=. ICLR (Poster) , year=

work page
[24]

International Conference on Learning Representations , year=

Denoising Diffusion Implicit Models , author=. International Conference on Learning Representations , year=

work page
[25]

Advances in Neural Information Processing Systems , volume=

Diffusion models beat gans on image synthesis , author=. Advances in Neural Information Processing Systems , volume=

work page
[26]

2009 , publisher=

Learning multiple layers of features from tiny images , author=. 2009 , publisher=

work page 2009
[27]

Advances in neural information processing systems , volume=

Improved techniques for training gans , author=. Advances in neural information processing systems , volume=

work page
[28]

Advances in neural information processing systems , volume=

Gans trained by a two time-scale update rule converge to a local nash equilibrium , author=. Advances in neural information processing systems , volume=

work page
[29]

ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models , year=

On Fast Sampling of Diffusion Probabilistic Models , author=. ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models , year=

work page
[30]

Advances in Neural Information Processing Systems , volume=

Maximum likelihood training of score-based diffusion models , author=. Advances in Neural Information Processing Systems , volume=

work page
[31]

Hierarchical Text-Conditional Image Generation with CLIP Latents

Hierarchical text-conditional image generation with clip latents , author=. arXiv preprint arXiv:2204.06125 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[32]

International Conference on Machine Learning , pages=

Improved denoising diffusion probabilistic models , author=. International Conference on Machine Learning , pages=. 2021 , organization=

work page 2021
[33]

International Conference on Learning Representations , year=

WaveGrad: Estimating Gradients for Waveform Generation , author=. International Conference on Learning Representations , year=

work page
[34]

International Conference on Learning Representations , year=

DiffWave: A Versatile Diffusion Model for Audio Synthesis , author=. International Conference on Learning Representations , year=

work page
[35]

Proceedings of the IEEE/CVF International Conference on Computer Vision , pages=

Score-based point cloud denoising , author=. Proceedings of the IEEE/CVF International Conference on Computer Vision , pages=

work page
[36]

Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages=

Diffusion probabilistic models for 3d point cloud generation , author=. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages=

work page
[37]

Proceedings of the IEEE/CVF International Conference on Computer Vision , pages=

3d shape generation and completion through point-voxel diffusion , author=. Proceedings of the IEEE/CVF International Conference on Computer Vision , pages=

work page
[38]

arXiv preprint arXiv:2112.07804 , year=

Tackling the Generative Learning Trilemma with Denoising Diffusion GANs , author=. arXiv preprint arXiv:2112.07804 , year=

work page arXiv
[39]

Journal of Machine Learning Research , volume=

Cascaded diffusion models for high fidelity image generation , author=. Journal of Machine Learning Research , volume=

work page
[40]

A measure-theoretical point of view , author=

Reciprocal processes. A measure-theoretical point of view , author=. Probability Surveys , volume=. 2014 , publisher=

work page 2014
[41]

IEEE transactions on visualization and computer graphics , volume=

The ball-pivoting algorithm for surface reconstruction , author=. IEEE transactions on visualization and computer graphics , volume=. 1999 , publisher=

work page 1999
[42]

ShapeNet: An Information-Rich 3D Model Repository

Shapenet: An information-rich 3d model repository , author=. arXiv preprint arXiv:1512.03012 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[43]

Non-Denoising Forward-Time Diffusions , author=

work page
[44]

Conference on Learning Theory , pages=

Theoretical guarantees for sampling and inference in generative models with latent diffusions , author=. Conference on Learning Theory , pages=. 2019 , organization=

work page 2019
[45]

Journal of Computational and Graphical Statistics , volume=

The art of data augmentation , author=. Journal of Computational and Graphical Statistics , volume=. 2001 , publisher=

work page 2001
[46]

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

Maximum likelihood from incomplete data via the EM algorithm , author=. Journal of the Royal Statistical Society: Series B (Methodological) , volume=. 1977 , publisher=

work page 1977
[47]

1984 , publisher=

Classical potential theory and its probabilistic counterpart , author=. 1984 , publisher=

work page 1984
[48]

The Girsanov theorem without (so much) stochastic analysis , author=. S. 2018 , publisher=

work page 2018
[49]

Mathematische Annalen , volume=

A generalization of Caffarelli’s contraction theorem via (reverse) heat flow , author=. Mathematische Annalen , volume=. 2012 , publisher=

work page 2012
[50]

Communications in Contemporary Mathematics , volume=

Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map , author=. Communications in Contemporary Mathematics , volume=. 2021 , publisher=

work page 2021
[51]

Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed

Knowledge distillation in iterative generative models for improved sampling speed , author=. arXiv preprint arXiv:2101.02388 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[52]

International Conference on Learning Representations , year=

Spectral Normalization for Generative Adversarial Networks , author=. International Conference on Learning Representations , year=

work page
[53]

International Conference on Learning Representations , year=

Refining Deep Generative Models via Discriminator Gradient Flow , author=. International Conference on Learning Representations , year=

work page
[54]

arXiv preprint arXiv:2202.09671 , year=

Truncated diffusion probabilistic models , author=. arXiv preprint arXiv:2202.09671 , year=

work page arXiv
[55]

International Conference on Learning Representations , year=

Tackling the Generative Learning Trilemma with Denoising Diffusion GANs , author=. International Conference on Learning Representations , year=

work page
[56]

Advances in Neural Information Processing Systems , volume=

Differentiable augmentation for data-efficient gan training , author=. Advances in Neural Information Processing Systems , volume=

work page
[57]

2009 , publisher=

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

work page 2009
[58]

2021 , publisher=

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

work page 2021
[59]

2021 , publisher=

Lectures on optimal transport , author=. 2021 , publisher=

work page 2021
[60]

Transport equations and multi-D hyperbolic conservation laws , pages=

Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields , author=. Transport equations and multi-D hyperbolic conservation laws , pages=. 2008 , publisher=

work page 2008
[61]

Advances in neural information processing systems , volume=

Generative adversarial nets , author=. Advances in neural information processing systems , volume=

work page
[62]

Auto-Encoding Variational Bayes

Auto-encoding variational bayes , author=. arXiv preprint arXiv:1312.6114 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[63]

Density estimation using Real NVP

Density estimation using real nvp , author=. arXiv preprint arXiv:1605.08803 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[64]

International conference on machine learning , pages=

Variational inference with normalizing flows , author=. International conference on machine learning , pages=. 2015 , organization=

work page 2015
[65]

Foundations and Trends

Computational optimal transport: With applications to data science , author=. Foundations and Trends. 2019 , publisher=

work page 2019
[66]

Optimal transport for applied mathematicians , author=. Birk. 2015 , publisher=

work page 2015
[67]

An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows , author=

work page
[68]

Communications on Pure and Applied Mathematics , volume=

Data-driven optimal transport , author=. Communications on Pure and Applied Mathematics , volume=. 2016 , publisher=

work page 2016
[69]

Advances in neural information processing systems , volume=

Neural ordinary differential equations , author=. Advances in neural information processing systems , volume=

work page
[70]

Advances in neural information processing systems , volume=

Glow: Generative flow with invertible 1x1 convolutions , author=. Advances in neural information processing systems , volume=

work page
[71]

Proceedings of the IEEE international conference on computer vision , pages=

Unpaired image-to-image translation using cycle-consistent adversarial networks , author=. Proceedings of the IEEE international conference on computer vision , pages=

work page
[72]

arXiv preprint arXiv:2202.07477 , year=

Understanding ddpm latent codes through optimal transport , author=. arXiv preprint arXiv:2202.07477 , year=

work page arXiv
[73]

Applied Mathematics Letters , pages=

The flow map of the Fokker--Planck equation does not provide optimal transport , author=. Applied Mathematics Letters , pages=. 2022 , publisher=

work page 2022
[74]

Stochastic analysis 2010 , pages=

Equivalence of stochastic equations and martingale problems , author=. Stochastic analysis 2010 , pages=. 2011 , publisher=

work page 2010
[75]

arXiv preprint arXiv:2208.14699 , year=

Let us Build Bridges: Understanding and Extending Diffusion Generative Models , author=. arXiv preprint arXiv:2208.14699 , year=

work page arXiv
[76]

preprint , year=

Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow , author=. preprint , year=

work page
[77]

International Conference on Machine Learning , pages=

Deep unsupervised learning using nonequilibrium thermodynamics , author=. International Conference on Machine Learning , pages=. 2015 , organization=

work page 2015
[78]

, author=

Normalizing Flows for Probabilistic Modeling and Inference. , author=. J. Mach. Learn. Res. , volume=

work page
[79]

Advances in neural information processing systems , volume=

Relative loss bounds for multidimensional regression problems , author=. Advances in neural information processing systems , volume=

work page
[80]

IEEE Transactions on Neural Networks , volume=

Relative loss bounds for single neurons , author=. IEEE Transactions on Neural Networks , volume=. 1999 , publisher=

work page 1999

Showing first 80 references.