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.
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Rectified Flow: A Marginal Preserving Approach to Optimal Transport
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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 $c$.
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representative citing papers
IFM learns deterministic tangent velocity fields on CP^{d-1} via Pancharatnam phase-aligned paths, recovering marginal transport with endpoint and stability guarantees while showing empirical gains over Euclidean flow matching on quantum benchmarks.
FAFM performs flow matching in the frequency domain using DCT on action sequences to produce continuous temporally consistent robotic actions with a Sobolev-style smoothness regularizer.
STREAM applies stochastic Riemannian flow matching on VFM-derived unit hypersphere latents with a novel anisotropic decoder to achieve SOTA reconstruction and generation on breast and colorectal cancer histopathology datasets.
Establishes a general lower bound of order ε^{1/(d+2)} on the localization rate of QOT optimizers around the Monge coupling in directed Hausdorff distance, with sharper affine-case tube bounds.
A control-theoretic linear program yields value-driven transport policies for generative modeling with straight paths and simulation-free training.
AsymFlow uses rank-asymmetric velocity prediction to reach 1.57 FID on ImageNet 256x256 and enables finetuning of latent flow models into superior pixel-space text-to-image generators.
Discrete MeanFlow parameterizes CTMC conditional transition kernels with a boundary-by-construction design to enable exact one-step generation in discrete state spaces.
A2A flow matching starts action generation from prior proprioceptive actions in latent space to enable single-step high-quality predictions in robotic policies.
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.
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 controlling kinetic energy tails.
FUSE adapts a multimodal diffusion transformer for flow-matching-based posterior estimation and adds inference-time likelihood-guided particle resampling, achieving state-of-the-art posterior fidelity on SBI benchmarks and real-world exoplanet orbital estimation.
ACID improves decision-time planning in world models by adding per-step action consistency residuals from an inverse dynamics model to the planning cost via an adaptive weight, yielding better performance with less compute across manipulation and navigation tasks.
LAFM adapts the source distribution in flow matching policies via a latent action model to better match fragmented robotic action spaces, claiming 23.4% higher real-world success and 10.4% on LIBERO-90 while beating larger pre-trained models.
SynthICL trains flow-matching transformer policies for in-context imitation learning entirely from synthetic RGB data and reports 79% average success on 16 unseen real manipulation tasks with one test-time demonstration.
OpenEAI-Platform delivers an open-source low-cost robotic arm and VLA model that outperforms commercial arms and matches large pretrained baselines on four real-world manipulation tasks using limited open data.
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 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 and vocabulary size m.
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.
Pretraining a vision-language model to output discrete 3D pose tokens on large non-robotic data, before training a robot action head, improves downstream manipulation success and data efficiency.
MCFlow uses decoupled flow time axes for atom types and crystal structures so a single model handles crystal structure prediction, de novo generation, and atom-type generation.
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 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.
π_{0.5} is a VLA model that achieves long-horizon dexterous manipulation in entirely new homes through co-training on heterogeneous tasks and multi-source data including web and semantic predictions.
citing papers explorer
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Building Normalizing Flows with Stochastic Interpolants
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.
-
Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport
IFM learns deterministic tangent velocity fields on CP^{d-1} via Pancharatnam phase-aligned paths, recovering marginal transport with endpoint and stability guarantees while showing empirical gains over Euclidean flow matching on quantum benchmarks.
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Frequency-Aware Flow Matching for Continuous and Consistent Robotic Action Generation
FAFM performs flow matching in the frequency domain using DCT on action sequences to produce continuous temporally consistent robotic actions with a Sobolev-style smoothness regularizer.
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STREAM: Stochastic Riemannian Flow Matching with Anisotropic Decoder for Digital Histopathology Image Generation
STREAM applies stochastic Riemannian flow matching on VFM-derived unit hypersphere latents with a novel anisotropic decoder to achieve SOTA reconstruction and generation on breast and colorectal cancer histopathology datasets.
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Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis
Establishes a general lower bound of order ε^{1/(d+2)} on the localization rate of QOT optimizers around the Monge coupling in directed Hausdorff distance, with sharper affine-case tube bounds.
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Generative Modeling by Value-Driven Transport
A control-theoretic linear program yields value-driven transport policies for generative modeling with straight paths and simulation-free training.
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Asymmetric Flow Models
AsymFlow uses rank-asymmetric velocity prediction to reach 1.57 FID on ImageNet 256x256 and enables finetuning of latent flow models into superior pixel-space text-to-image generators.
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Discrete MeanFlow: One-Step Generation via Conditional Transition Kernels
Discrete MeanFlow parameterizes CTMC conditional transition kernels with a boundary-by-construction design to enable exact one-step generation in discrete state spaces.
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Action-to-Action Flow Matching
A2A flow matching starts action generation from prior proprioceptive actions in latent space to enable single-step high-quality predictions in robotic policies.
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Generative Modeling of Discrete Data Using Geometric Latent Subspaces
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.
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On The Hidden Biases of Flow Matching Samplers
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 controlling kinetic energy tails.
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FUSE: FK-Steered Multi-Modal Flow Matching for Efficient Simulation-Based Posterior Estimation
FUSE adapts a multimodal diffusion transformer for flow-matching-based posterior estimation and adds inference-time likelihood-guided particle resampling, achieving state-of-the-art posterior fidelity on SBI benchmarks and real-world exoplanet orbital estimation.
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ACID: Action Consistency via Inverse Dynamics for Planning with World Models
ACID improves decision-time planning in world models by adding per-step action consistency residuals from an inverse dynamics model to the planning cost via an adaptive weight, yielding better performance with less compute across manipulation and navigation tasks.
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Flowing With Purpose: Latent Action Guided Flow Matching Policies For Robotic Manipulation
LAFM adapts the source distribution in flow matching policies via a latent action model to better match fragmented robotic action spaces, claiming 23.4% higher real-world success and 10.4% on LIBERO-90 while beating larger pre-trained models.
-
SynthICL: Scalable In-context Imitation Learning with Synthetic Data
SynthICL trains flow-matching transformer policies for in-context imitation learning entirely from synthetic RGB data and reports 79% average success on 16 unseen real manipulation tasks with one test-time demonstration.
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OpenEAI-Platform: An Open-source Embodied Artificial Intelligence Hardware-Software Unified Platform
OpenEAI-Platform delivers an open-source low-cost robotic arm and VLA model that outperforms commercial arms and matches large pretrained baselines on four real-world manipulation tasks using limited open data.
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Discrete Flow Matching for Offline-to-Online Reinforcement Learning
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.
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Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions
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 and vocabulary size m.
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Efficient Diffusion Distillation via Embedding Loss
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.
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PoseVLA: Universal Pose Pretraining for Generalizable Vision-Language-Action Policies
Pretraining a vision-language model to output discrete 3D pose tokens on large non-robotic data, before training a robot action head, improves downstream manipulation success and data efficiency.
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Multimodal Crystal Flow: Any-to-Any Modality Generation for Unified Crystal Modeling
MCFlow uses decoupled flow time axes for atom types and crystal structures so a single model handles crystal structure prediction, de novo generation, and atom-type generation.
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SPEAR-1: Scaling Beyond Robot Demonstrations via 3D Understanding
SPEAR-1 combines a 3D-enriched VLM with embodied control to match or exceed existing robotic foundation models using 20 times fewer robot demonstrations.
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SmolVLA: A Vision-Language-Action Model for Affordable and Efficient Robotics
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.
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$\pi_{0.5}$: a Vision-Language-Action Model with Open-World Generalization
π_{0.5} is a VLA model that achieves long-horizon dexterous manipulation in entirely new homes through co-training on heterogeneous tasks and multi-source data including web and semantic predictions.
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Regional climate risk assessment from climate models using probabilistic machine learning
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$\pi_0$: A Vision-Language-Action Flow Model for General Robot Control
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.
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Wavelet-Guided Semantic Signal Compensation for Inversion-Free Image Editing
Proposes a frequency-aware semantic compensation strategy using wavelets to strengthen text-conditioned signals in early diffusion steps for better global editing without inversion.
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C$^2$FG: Control Classifier-Free Guidance via Score Discrepancy Analysis
C²FG provides a time-dependent guidance controller for diffusion models derived from score discrepancy upper bounds, implemented as an exponential decay function without retraining.
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GR-3 Technical Report
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.
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On the Power of Foundation Models
Category theory proves prompt-based learning on perfect foundation models works only for representable tasks, fine-tuning solves tasks in the pretext category, and models can represent unseen target-category objects using source-category structure.
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OmniVLA-RL: A Vision-Language-Action Model with Spatial Understanding and Online RL
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.
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Generative models for decision-making under distributional shift
Flow- and score-based generative models can construct nominal, stressed, and conditional distributions for decision-making under distributional shift via transport maps and probability-space optimization.
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Stabilization Learning: A Paradigm Transition Bridging Control Theory and Machine Learning
Stabilization learning is introduced as a stability-centric framework bridging control theory and machine learning via a six-tuple mathematical model applicable to control, observation, and recognition tasks.
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Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans
A mathematical review of flow matching techniques for generative models, showing characterizations via couplings, kernels, and processes, with application to inverse problems.