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
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.
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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.
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.
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.
Pose-VLA uses a decoupled two-stage pre-training with discrete pose tokens to extract universal 3D spatial priors from 3D datasets and robotic trajectories, achieving 79.5% success on RoboTwin 2.0 and 96.0% on LIBERO.
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.
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GenFocal uses probabilistic ML to downscale coarse climate projections to fine-scale weather events without paired training data and samples rare high-impact events more accurately than prior methods.
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